Specific Heat Capacity

Specific heat capacity

Specific heat capacity of a substance is defined as the quantity of heat required to raise the temperature of 1 kg of the substance through 1K. Its unit is J kg^–1K^–1.

Molar specific heat capacity of a gas

Molar specific heat capacity of a gas is defined as the quantity of heat required to raise the temperature of 1 mole of the gas through 1K. Its unit is J mol^–1 K^–1.

Specific heat capacity of a gas may have any value between –∞ and +∞ depending upon the way in which heat energy is given.

Let m be the mass of a gas and C its specific heat capacity. Then ∆Q = m × C × ∆T where ∆Q is the amount of heat absorbed and ∆T is the corresponding rise in temperature.

That is,

Case (i)

If the gas is insulated from its surroundings and is suddenly compressed, it will be heated up and there is rise in temperature, even though no heat is supplied from outside

(i.e)   ∆Q = 0

∴      C   = 0

Case (ii)

If the gas is allowed to expand slowly, in order to keep the temperature constant, an amount of heat ∆Q is supplied from outside,

Then, C = ∆Q/m×∆T = ∆Q/0 = +∞

(∵ ∆Q is +ve as heat is supplied from outside)

Case (iii)

If the gas is compressed gradually and the heat generated ∆Q is conducted away so that temperature remains constant, then

Then, C = – ∆Q/m×∆T = – ∆Q/0 = – ∞

(∵ ∆Q is – ve as heat is supplied by the system)

Thus we find that if the external conditions are not controlled, the value of the specific heat capacity of a gas may vary from +∞ to -∞

Hence, in order to find the value of specific heat capacity of a gas, either the pressure or the volume of the gas should be kept constant.

Consequently a gas has two specific heat capacities (i) Specific heat capacity at constant volume (ii) Specific heat capacity at constant pressure.

Molar specific heat capacity of a gas at constant volume

Molar specific heat capacity of a gas at constant volume C_V is defined as the quantity of heat required to raise the temperature of one mole of a gas through 1 K, keeping its volume constant.

Molar specific heat capacity of a gas at constant pressure

Molar specific heat capacity of a gas at constant pressure Cp is defined as the quantity of heat to raise the temperature of one mole of a gas through 1 K keeping its pressure constant.

Specific heat capacity of monoatomic, diatomic and triatomic gases

Monoatomic gases like argon, helium etc. have three degrees of freedom.

We know, kinetic energy per molecule, per degrees of freedom is ½ kT.

Thus, kinetic energy per molecule with three degrees of freedom is 3/2 kT.

Total kinetic energy of one mole of the monoatomic gas is given by E = (3/2 kT) N = 3/2 RT, where N is the Avogadro number.

So, dE/dT = 3/2 R

If dE is a small amount of heat required to raise the temperature of 1 mole of the gas at constant volume, through a temperature dT,

dE = (1) (C_v) (dT)

C_v = dE/dT = 3/2 R

As R = 8.31 J mol^-1 K^-1

C_v = 3/2 ( 8.31) = 12.465 J mol^-1 K^-1

Then C_P – C_V = R

C_P = C_V + R = 3/2 R + R = 5/2 R = (5/2) (8.31)

So, C_P = 20.775 J mol^-1 K^-1

In diatomic gases like hydrogen, oxygen, nitrogen etc., a molecule has five degrees of freedom. Hence the total energy associated with one mole of diatomic gas is,

E = (5) (1/2 kT) (N) = 5/2 (RT)

Also, C_v = dE/dT = d/dT [5/2 (RT)] = 5/2 R

C_v = (5/2) (8.31) = 20.775 J mol^-1 K^-1

But, C_P = C_v + R = 5/2 R + R = 7/2 R

C_P = (7/2) (8.31) = 29.085 J mol^-1 K^-1

Similarly, C_P and C_V can be calculated for triatomic gases.

Internal energy

Internal energy U of a system is the energy possessed by the system due to molecular motion and molecular configuration. The internal kinetic energy UK of the molecules is due to the molecular motion and the internal potential energy U_P is due to molecular configuration. Thus

U = U_K + U_P

It depends only on the initial and final states of the system. In case of an ideal gas, it is assumed that the intermolecular forces are zero. Therefore, no work is done, although there is change in the intermolecular distance. Thus U_P = O. Hence, internal energy of an ideal gas has only internal kinetic energy, which depends only on the temperature of the gas.

In a real gas, intermolecular forces are not zero. Therefore, a definite amount of work has to be done in changing the distance between the molecules. Thus the internal energy of a real gas is the sum of internal kinetic energy and internal potential energy. Hence, it would depend upon both the temperature and the volume of the gas.

Refer this video to know more about on, “Heat Capacity & Specific Heat Capacity”.

If specific heat capacity is constant, the temperature will rise at a uniform rate so long as the power input is constant and no energy is lost to the outside. There are large potential heat losses if the substance is not well insulated. These can be accounted for but in most cases students will not do so quantitatively. They should calculate their value and make a comparison with data book values. They should be able to think of a number of reasons why their value does not match that in the data book. Unlike the total heat capacity, the specific heat capacity is independent of mass or volume. It describes how much heat must be added to a unit of mass of a given substance to raise its temperature by one degree Celsius. The units of specific heat capacity are J/(kg °C) or equivalently J/(kg K). The heat capacity and the specific heat are related by C=cm or c=C/m. The mass m, specific heat c, change in temperature ΔT, and heat added (or subtracted) Q are related by the equation: Q=mcΔT. Values of specific heat are dependent on the properties and phase of a given substance. Since they cannot be calculated easily, they are empirically measured and available for reference in tables. The specific heat is the amount of heat per unit mass required to raise the temperature by one degree Celsius. The heat capacity of a substance is the amount of heat energy it must consume in order to raise its temperature by 1K or 1ºC; it can be expressed using either the units joules or calories. The specific heat (also called specific heat capacity) is the amount of heat required to change a unit mass (or unit quantity, such as mole) of a substance by one degree in temperature. Therefore, unlike the extensive variable heat capacity, which depends on the quantity of material, specific heat is an intensive variable and has units of energy per mass per degree (or energy per number of moles per degree).

Problem (JEE Advanced):

A gas is a mixture of 2 moles of oxygen and 4 moles of argon at temperature T. Neglecting vibrational modes, show that the energy of the system is 11 RT where R is the universal gas constant.

Solution:

Since oxygen is a diatomic moleucle with 5 degrees of freedom, degrees of freedom of molecules in 2 moles of oxygen

= f1 = 2 N × 5 = 10 N

Since argon is a monatomic molecules degrees of freedom of molecules in 4 moles of argon = f2 = 4 N × 3 = 12 N

Total degrees of freedom of the mixture = f = f₁ + f₂ = 22 N

As per the principle of law of equipartition of energy, energy associated with each degree of freedom of a molecule = ½ kT

Total energy of the system = ½ kT × 22 N = 11 RT 

The specific heat capacity of a substance is equal to

(a) mass of the substance × heat capacity

(b) heat capacity ⁄ mass of the substance

(c) mass of the substance ⁄ heat capacity

(d) None of the above

Question 2

In the science lab, a student heats up a chemical from 10 °C to 25 °C which requires the thermal energy of 30000 J. If the mass of the object is 40 kg, the specific heat capacity of the chemical would be

(a) 25 J kg^-1 °C^-1

(b) 50 J kg^-1 °C^-1

(c) 75 J kg^-1 °C^-1

(d) 100 J kg^-1 °C^-1

Question 3

The specific heat capacity of a substance is equal to

(a) the amount of heat required to raise the temperature of a 1 kg of a substance by 1 K

(b) the amount of heat required to raise the temperature of a substance by 1 K

(c) the amount of heat required to change the phase of a substance from solid to liquid without any change

(d) the amount of heat required to change the phase of a substance from liquid to gas without any change

Question 4

The capacity to do work is called as:

(a) Heat

(b) Energy

(c) Work

(d) None of above

Question 5

Heat is measured in

(a) Joule

(b) Calorie

(c) Both A and B

(d) Joule/second

Q.1	Q.2	Q.3	Q.4	Q.5
b	b	a	b	a

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