Real Gases | Definition, Formula, Units – Kinetic Theory of Gases

Real Gases Definition:
Real gases are nonideal gases whose molecules occupy space and have interactions; consequently, they do not adhere to the ideal gas law.
Real gases deviate slightly from ideal gas laws because

  1. Real gas molecules attract one another.
  2. Real gas molecules occupy a finite volume.

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Real Gases | Definition, Formula, Units – Kinetic Theory of Gases

Real or van der Waals’ Gas Equation
\(\left(p+\frac{a}{V^{2}}\right)\) (V – b) = RT

where, a and b are called van der Waals’ constants.

Dimension [a] = [ML5T-2] and [b] = [L3]
Units a = N-m4 and b = m3.

Note:
Real gases obey this equation at high pressure and low temperature

Pressure of a gas
Pressure due to an ideal gas is given by
p = \(\frac{1}{3} \frac{m n}{V}\)v2 = \(\frac{1}{3} \rho \bar{v}^{2}\)

For one mole of an ideal gas, where, m = mass of one molecule, n = number of molecules,

V = volume of gas, \(\bar{v}=\sqrt{\frac{\bar{v}_{1}^{2}+\bar{v}_{2}^{2}+\ldots+\bar{v}_{n}^{2}}{n}}\)
is called root mean square (rms) velocity of the gas molecules and M = molecular weight of the gas.

If p is the pressure of the gas and E is the kinetic energy per unit volume is E, then
p = \(\frac{2}{3}\)E

Note:
Effect of mass, volume and temperature on pressure

  • when volume and temperature of a gas are constant, then pressure ∝ mass of gas.
  • when mass and temperature of a gas are constant, then pressure ∝ \(\frac{1}{\text { volume }}\)
  • when mass and volume of gas are constant, then pressure ∝ temperature ∝ c2.

Kinetic Energy of a Gas and Speed of Gas Molecules

(i) Average kinetic energy of translation per molecule of a gas is given by

E = \(\frac{3}{2}\) kT

where, k = Boltzmann’s constant.

(ii) Average kinetic energy of translation per mole of a gas is given by

E = \(\frac{3}{2}\) RT

where, R = universal gas constant.

(iii) For a given gas kinetic energy

E ∝ T
⇒ \(\frac{E_{1}}{E_{2}}=\frac{T_{1}}{T_{2}}\)

(iv) Root mean square (rms) velocity of the gas molecules is given by

\(v\) = \(\sqrt{\frac{3 R T}{M}}=\sqrt{\frac{3 p}{\rho}}\)

(v) For a given gas, \(v\) ∝ \(\sqrt{T}\)

(vi) For different gases, \(v\) ∝ \(\frac{1}{\sqrt{M}}\)

(vii) Boltzmann’s constant, k = \(\frac{R}{N}\)
where, R is an ideal gas constant and N = Avogadro number.
Value of Boltzmann’s constant is 1.38 × 10-28 J/K.

(viii) The average speed of molecules of a gas is given by

\(\bar{v}=\sqrt{\frac{8 k T}{\pi m}}=\sqrt{\frac{8 R T}{\pi M}}\)

(ix) The most probable speed of molecules u of a gas is given by

\(v_{\mathrm{mp}}=\sqrt{\frac{2 k T}{m}}=\sqrt{\frac{2 R T}{M}} \Rightarrow v_{\mathrm{rms}}>\bar{v}>v_{\mathrm{mp}}\)

(x) With rise in temperature rms speed of gas molecules increases as

\(v_{\mathrm{rms}}\) ∝ \(\sqrt{T}\)

(xi) With the increase in molecular weight rms speed of gas molecule decrease as

\(v_{\mathrm{rms}}\) ∝ \(\frac{1}{\sqrt{M}}\)

(xii) Rms speed of gas molecules is of the order of km/s, e.g. at NTP for hydrogen gas

\(v_{\mathrm{rms}}\) = \(\sqrt{\frac{3 R T}{M}}=\sqrt{\frac{3 \times 8.31 \times 273}{2 \times 10^{3}}}\) = 1.84

(xiii) Rms speed of gas molecules does not depend on the pressure of gas (if temperature remains constant) because p ∝ ρ (Boyle’s law). If pressure is increased n times, then density will also increase by n times but υrms remains constant.

Gas Laws in Physics | Boyle’s Law, Charles’ Law, Gay Lussac’s Law, Avogadro’s Law – Kinetic Theory of Gases

Gas Laws physics:
Through experiments, it was established that gases irrespective of their nature obey the following laws

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Gas Laws in Physics | Boyle’s Law, Charles’ Law, Gay Lussac’s Law, Avogadro’s Law – Kinetic Theory of Gases

Boyle’s Law is represented by the equation:
At constant temperature, the volume (V) of given mass of a gas is inversely proportional to its pressure (p), i.e.
Gas Laws

V ∝ \(\frac{1}{p}\) ⇒ pV = constant

For a given gas, p1V1 = p2V2

Charles’ Law
At constant pressure, the volume (V) of a given mass of gas is directly proportional to its absolute temperature (T), i.e.

V ∝T ⇒ \(\frac{V}{T}\) = constant

For a given gas, \(\frac{V_{1}}{T_{1}}=\frac{V_{2}}{T_{2}}\)

At constant pressure, the volume (V) of a given mass of a gas increases or decreases \(\frac{1}{273.15}\) by of its volume at 0°C for each 1°C rise or fall in temperature.
Image
Volume of the gas at t°C,
Vt = V0\(\left(1+\frac{t}{273.15}\right)\)

where, V0 is the volume of gas at 0°C.

Gay Lussac’s or Regnault’s Law
At constant volume, the pressure p of a given mass of gas is directly proportional to its absolute temperature T, i.e.

p ∝ T ⇒ \(\frac{P}{T}\) = constant

For a given gas, \(\frac{p_{1}}{T_{1}}=\frac{p_{2}}{T_{2}}\)

At constant volume, the pressure p of a given mass of a gas increases or decreases by \(\left(1+\frac{t}{273.15}\right)\) of its pressure at 0°C for each 1°C rise or fall in
Gas Laws

Volume of the gas at t°C,
pt = p0\(\left(1+\frac{t}{273.15}\right)\)

where, p0 is the pressure of gas at 0°C.

Avogadro’s Law
Avogadro stated that equal volume of all the gases under similar conditions of temperature and pressure contain equal number of molecules. This statement is called Avogadro’s hypothesis. According to Avogadro’s law N1 = N2, where N2 and N2 are number of molecules in two gases respectively.

(i) Avogadro’s number:
The number of molecules present in lg mole of a gas is defined as Avogadro’s number.
NA = 6.023 x 1023 per gram mole

(ii) At STP or NTP (T = 273 K and p = 1 atm), 22.4 L of each gas has 6.023 x 1023 molecules.

(iii) One mole of any gas at STP occupies 22.4 L of volume.

Dalton’s Law of Partial Pressure
It states that the total pressure of a mixture of non-interacting ideal gases is the sum of partial pressures exerted by individual gases in the mixture, i.e. p = p1 + p2 + p3 + ………

What is Triple Point of Water? | Definition – Thermometry and Calorimetry

Triple Point of Water Definition:
The values of pressure and temperature at which water coexists in equilibrium in all three states of matter, i.e. ice, water and vapour is called triple point of water.

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What is Triple Point of Water? | Definition – Thermometry and Calorimetry

Triple point of water is 273 K temperature and 0.46 cm of mercury pressure.

The triple point of water, T3 = 273.16 K, is the standard fixed-point temperature for the calibration of thermometers. This agreement also sets the size of the kelvin as 1/273.16 of the difference between the triple-point temperature of water and absolute zero.

Why is the Triple Point of Water Important

It is also important to note that the triple point of water correlates with the pressure necessary for liquid water to exist. Because ice is less dense than liquid water, ice frozen at pressures below the triple point will sublime directly into water vapor.

Thermometry and Calorimetry:
The thermometer is a device used to check the temperature of an object. This branch of measurement of the temperature of a substance is called thermometry. It is measured in degrees or Fahrenheit, usually.

Calorimetry also means the measurement of heat but in joules. It states the amount of heat lost by the body is the amount of heat gained by its surrounding.

Heat Energy Temperature and its Measurement
Thermometric Property Thermometers
Thermal Expansion Thermal Equilibrium
Triple Point of Water Specific Heat Capacity
Thermal Capacity Water Equivalent
Latent Heat Joule’s Law
Calorimetry

What is Water Equivalent? | Definition, Symbol, Units, Formula – Thermometry and Calorimetry

Water Equivalent Definition:
It is the quantity of water whose thermal capacity is same as the heat capacity of the body. Water equivalent is generally used in comparison of a water quantity.

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What is Water Equivalent? | Definition, Symbol, Units, Formula – Thermometry and Calorimetry

Water Equivalent Symbol:
It is denoted by W.

W = ms = heat capacity of the body.

Water Equivalent Unit:
SI unit for water is kg
Its expressed in the unit gram.

Water Equivalent Dimensional Formula:
Dimensional formula is [M1L0T0].

Thermometry and Calorimetry:
The thermometer is a device used to check the temperature of an object. This branch of measurement of the temperature of a substance is called thermometry. It is measured in degrees or Fahrenheit, usually.

Calorimetry also means the measurement of heat but in joules. It states the amount of heat lost by the body is the amount of heat gained by its surrounding.

Heat Energy Temperature and its Measurement
Thermometric Property Thermometers
Thermal Expansion Thermal Equilibrium
Triple Point of Water Specific Heat Capacity
Thermal Capacity Water Equivalent
Latent Heat Joule’s Law
Calorimetry

Thermal Equilibrium | Definition, Examples – Thermometry and Calorimetry

Thermal Equilibrium Definition Physics:
When there is no transfer of heat between two bodies in contact, then the bodies are called in thermal equilibrium. The objects in thermal equilibrium have the same temperature.

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Thermal Equilibrium | Definition, Examples – Thermometry and Calorimetry

Thermal Equilibrium Examples:

  • The cooling object in the refrigerator
  • Taking the temperature of a sick patient (Thermometer)
  • Rice cooker with a thermostat
  • Baking cake in an oven
  • Cooling Drinks
  • A wet towel is placed on the forehead of a person who has high fever.

Thermometry and Calorimetry:
The thermometer is a device used to check the temperature of an object. This branch of measurement of the temperature of a substance is called thermometry. It is measured in degrees or Fahrenheit, usually.

Calorimetry also means the measurement of heat but in joules. It states the amount of heat lost by the body is the amount of heat gained by its surrounding.

Heat Energy Temperature and its Measurement
Thermometric Property Thermometers
Thermal Expansion Thermal Equilibrium
Triple Point of Water Specific Heat Capacity
Thermal Capacity Water Equivalent
Latent Heat Joule’s Law
Calorimetry

Principle of Calorimetry | Definition – Thermometry and Calorimetry

Calorimetry Physics Meaning:
This is the branch of heat transfer that deals with the measurement of heat. The heat is usually measured in calories or kilo calories.

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Principle of Calorimetry | Definition – Thermometry and Calorimetry

Calorimetry is the science or act of measuring changes in state variables of a body for the purpose of deriving the heat transfer associated with changes of its state.

Principle of Calorimetry:
When a hot body is mixed with a cold body, then heat lost by hot body is equal to the heat gained by cold body.

Heat lost = Heat gain

i.e. principle of calorimetry follows the law of conservation of heat energy.

If two substances having masses m1 and m2, specific heats c1, and c2 kept at temperatures T1 and T2 (T1 >T2) are mixed, such that temperature of mixture at equilibrium is Tmix.
Then,
m1.c1(T1 – Tmix) = m2c2(Tmix – T2)

or

Tmix = \(\frac{m_{1} c_{1} T_{1}+m_{2} c_{2} T_{2}}{m_{1} c_{1}+m_{2} c_{2}}\)

Temperature of Mixture in Different Cases

Calorimetry

Thermometry and Calorimetry:
The thermometer is a device used to check the temperature of an object. This branch of measurement of the temperature of a substance is called thermometry. It is measured in degrees or Fahrenheit, usually.

Calorimetry also means the measurement of heat but in joules. It states the amount of heat lost by the body is the amount of heat gained by its surrounding.

Heat Energy Temperature and its Measurement
Thermometric Property Thermometers
Thermal Expansion Thermal Equilibrium
Triple Point of Water Specific Heat Capacity
Thermal Capacity Water Equivalent
Latent Heat Joule’s Law
Calorimetry

Joule’s Law of Heating | Definition of Joule’s Law – Thermometry and Calorimetry

Joule’s Law 
According to Joule, whenever heat is converted into work or work is converted into heat, then the ratio of between work and heat in constant.

\(\frac{W}{Q}\) = J

where J is the mechanical equivalent of heat.

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Joule’s Law of Heating | Definition of Joule’s Law – Thermometry and Calorimetry

When water falls from height h, then increase in temperature dT at the bottom is

dT = \(\left(\frac{g h}{J \cdot C}\right)^{\circ} \mathrm{C}\)

  • When m kg of ice-block falls from height h and its some part m’ is melt down, then

h = \(\frac{m^{\prime}}{m}\left(\frac{J L}{g}\right)\) meter

If ice-block melts completely, then m = m’ and hence h = \(\frac{J L}{g}\)

Joule’s first law” (Joule heating), a physical law expressing the relationship between the heat generated and current flowing through a conductor.

Joule’s second law” states that the internal energy of an ideal gas is independent of its volume and pressure, depending only on its temperature.

Melting
Conversion of solid into liquid state at constant temperature is called melting.

Fusion and Freezing Point
The process of change of state from liquid to solid is called fusion. The temperature at which liquid starts to freeze is known as the freezing point of the liquid.

Evaporation
Conversion of liquid into vapour at all temperatures (even below its boiling point) is called evaporation.

Boiling
When a liquid is heated gradually, at a particular temperature the saturated vapour pressure of the liquid becomes equal to the atmospheric pressure, now bubbles of vapour rise to the surface of the liquid. This process is called boiling of the liquid.

The temperature at which liquid boils is called boiling point.

The boiling point of water increases with increase in pressure and decreases with decrease in pressure.

Sublimation
The conversion of a solid into vapour state is called sublimation.

Hoar Frost
The conversion of vapours into solid state is called hoar frost.

Thermometry and Calorimetry:
The thermometer is a device used to check the temperature of an object. This branch of measurement of the temperature of a substance is called thermometry. It is measured in degrees or Fahrenheit, usually.

Calorimetry also means the measurement of heat but in joules. It states the amount of heat lost by the body is the amount of heat gained by its surrounding.

Heat Energy Temperature and its Measurement
Thermometric Property Thermometers
Thermal Expansion Thermal Equilibrium
Triple Point of Water Specific Heat Capacity
Thermal Capacity Water Equivalent
Latent Heat Joule’s Law
Calorimetry

Latent Heat | Definition, Formula, Units – Thermometry and Calorimetry

Latent Heat Definition: (Change of State)
The heat energy absorbed or released at constant temperature per unit mass for change of state is called latent heat.

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Latent Heat | Definition, Formula, Units – Thermometry and Calorimetry

Latent Heat Formula Physics:
Heat energy absorbed or released during change of state is given by

Q = mL

where,
m = mass of the substance and
L = latent heat

Latent Heat Unit:
Its unit is cal/g or J/kg

Latent Heat Dimensional Formula:
Dimensional formula is [L2 T-2].

Latent Heat of Vaporization of Water:
For water at its normal boiling point or condensation temperature (100°C), the latent heat of vaporization is

L = 540 cal/g = 40.8 kJ/ mol
= 2260 kJ/kg

Latent Heat of Fusion of Water:
For water at its normal freezing temperature or melting point (0°C), the latent heat of fusion is

L = 80 cal/ g = 60 kJ/mol
= 336 kJ/kg

It is more painful to get burnt by steam rather than by boiling water at 100°C. Steam converted to water at 100°C, then it gives out 536 cal of heat, so, it is clear that steam at 100°C has more heat than water at 100°C (i.e. boiling of water).

After snow falls, the temperature of the atmosphere becomes very low. This is because the snow absorbs the heat from the atmosphere to melt down. So, in the mountains, when snow falls, one does not feel too cold but when ice melts, he feels too cold.

There is more shivering effect of icecream on teeth as compared to that of water (obtained from ice). This is because when icecream melts down, it absorbs large amount of heat from teeth.

Thermometry and Calorimetry:
The thermometer is a device used to check the temperature of an object. This branch of measurement of the temperature of a substance is called thermometry. It is measured in degrees or Fahrenheit, usually.

Calorimetry also means the measurement of heat but in joules. It states the amount of heat lost by the body is the amount of heat gained by its surrounding.

Heat Energy Temperature and its Measurement
Thermometric Property Thermometers
Thermal Expansion Thermal Equilibrium
Triple Point of Water Specific Heat Capacity
Thermal Capacity Water Equivalent
Latent Heat Joule’s Law
Calorimetry

Thermal (Heat) Capacity | Definition, Formula, Units – Thermometry and Calorimetry

Thermal Capacity Definition Physics: (Heat) Capacity
Heat capacity or thermal capacity is a physical property of matter, defined as the amount of heat to be supplied to a given mass of a material to produce a unit change in its temperature.

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Thermal (Heat) Capacity | Definition, Formula, Units – Thermometry and Calorimetry

Thermal Capacity Formula:
Heat capacity of any body is equal to the amount of heat energy required to increase its temperature through 1°C.

Heat capacity = mc

where, c = specific heat of the substance of the body and m = mass of the body.

Thermal Capacity Unit:
Its SI unit is joule/kelvin (J/K)

Thermal Capacity Dimensional Formula:
Dimensional formula [ML2T-2K-1].

Molar Specific Heat Capacity Formula:
Molar specific heat capacity,

c = \(\frac{s}{\mu}=\frac{\Delta Q}{\mu \Delta T}\)

where, µ = number of moles of substances (gas).

The relation between c and M are

c = MS

where,
M = molecular mass of substance and
S = specific heat capacity.

Thermometry and Calorimetry:
The thermometer is a device used to check the temperature of an object. This branch of measurement of the temperature of a substance is called thermometry. It is measured in degrees or Fahrenheit, usually.

Calorimetry also means the measurement of heat but in joules. It states the amount of heat lost by the body is the amount of heat gained by its surrounding.

Heat Energy Temperature and its Measurement
Thermometric Property Thermometers
Thermal Expansion Thermal Equilibrium
Triple Point of Water Specific Heat Capacity
Thermal Capacity Water Equivalent
Latent Heat Joule’s Law
Calorimetry

What is Specific Heat Capacity? | Definition, Units, Types – Thermometry and Calorimetry

Specific Heat Definition:
The amount of heat required to raise the temperature of unit mass of the substance through 1°C is called its specific heat.

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What is Specific Heat Capacity? | Definition, Units, Types – Thermometry and Calorimetry

Specific Heat Symbol:
It is denoted by c or s.

Specific Heat Unit:
Its SI unit is ‘joule/kilogram-°C’ (J/kg-°C) or Jkg-1K-1

Dimensional Formula of Specific Heat:
The dimensional formula is [L2T2θ-1].

Specific Heat of Water:
The specific heat of water is 4200 J kg-1°C-1 or 1 cal g-1 C-1 which is high as compared with most other substances.

Specific Heat Capacity Formula:
The amount of heat energy required to change the temperature of any substance is given by

Q = mcΔt

where,
m = mass of the substance,
c = specific heat of the substance and
Δt = change in temperature

Gases have two types of specific heat

  1. The specific heat capacity at constant volume (Cv).
  2. The specific heat capacity at constant pressure (Cp).

Specific heat at constant pressure (Cp) is greater than specific heat at constant volume (Cv), i.e. Cp>Cv.
For molar specific heats, Cp – Cv = R
where, R = gas constant and this relation is called Mayer’s formula.

The ratio of two principal sepecific heats of a gas is represented by γ, i.e.

γ = \(\frac{C _{p}}{C_{V}}\)

The value of γ depends on atomicity of the gas.

Thermometry and Calorimetry:
The thermometer is a device used to check the temperature of an object. This branch of measurement of the temperature of a substance is called thermometry. It is measured in degrees or Fahrenheit, usually.

Calorimetry also means the measurement of heat but in joules. It states the amount of heat lost by the body is the amount of heat gained by its surrounding.

Heat Energy Temperature and its Measurement
Thermometric Property Thermometers
Thermal Expansion Thermal Equilibrium
Triple Point of Water Specific Heat Capacity
Thermal Capacity Water Equivalent
Latent Heat Joule’s Law
Calorimetry