Specific Heat

 

Molar specific heat

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In many instances, the most convenient unit for specifying the amount of a substance is the mole, where

1 mole = 6.023 x 10²³ elementary units of any substance.

Thus, 1 mole of aluminum means 6.023 x 10²³ atoms (the atom being the elementary unit), and 1 mol of aluminum oxide means 6.023 x 10²³ molecules of the oxide (because the molecule is the elementary unit of a compound).

When quantities are expressed in moles, specific heats must also involve moles (rather than a mass unit). It is then called molar specific heat. The above table shows the values for some elemental solids (each consisting of a single element) at room temperature.

Specific heat of a gas

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The specific heat of a substance is the amount of heat required to increase the temperature of a unit mass of it through a unit temperature. Its units in cgs system are cal g^{-1 o}C^-1. In mks system,  it is measured in kilocal kg^{-1 o}C^-1. The SI unit is J kg^-1 K^-1.

The above definition is based on the assumption that the heat supplied to the substance, causes only a rise in the temperature of the substance. This assumption is valid only if the substance is heated at constant volume. However, a substance generally expands when heated. In that case, the heat supplied is utilized in two ways. A part of the heat supplied, is used to do mechanical work in moving the molecules apart, against forces of attraction between them and in expanding against atmospheric pressure. The rest of the heat supplied, increases the temperature of the substance.

In the case of solids and liquids, the coefficient of expansion is very small. So, the heat supplied is assumed to increase only the temperature. However, the coefficient of expansion is very large in case of gases. Thus, if a gas is heated the heat energy is required, not only to increase the temperature of the gas but also to do mechanical work in overcoming external pressure during expansion. In the case of gases, only a negligible amount of mechanical work is required to pull the molecules apart because the intermolecular forces are extremely weak.

Limits of specific heat of a gas

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Let us consider mass m of a gas. Let Q units of heat raise the temperature of the gas through DT. Then the specific heat of the gas  is given by

Consider a gas enclosed in a cylinder which is fitted with an airtight and frictionless piston.

(i) Let the gas be suddenly compressed. In this case, no heat is supplied to the gas. However, there is an increase in the temperature of the gas.

(ii) Let the gas be heated and allowed to expand. Suppose the 'fall in temperature due to expansion' is equal to the rise in temperature due to heat supplied.

(iii) Let the gas be heated and allowed to expand. Suppose, in this case, the 'fall in temperature due to expansion' is less than the 'rise in temperature due to heat supplied', the net effect is a rise in the temperature of the gas. Therefore DT is positive. Thus is positive.

(iv) Let the gas be heated and allowed to expand such that the 'fall in temperature due to expansion' is more than the 'rise in temperature due to heat supplied'. The net effect is a decrease in temperature of the gas. Therefore DT is negative. Thus, is negative.

We can conclude from the above examples that a gas does not possess a unique or a single specific heat. The specific heat of a gas may have any positive or negative value ranging from zero to infinity. The specific heat of a gas depends upon the manner in which it is heated. Thus, it is meaningless to talk about the specific heat of a gas unless the condition under which it is heated, is mentioned.

Specific heat of a gas at constant volume and pressure

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The following are of special significance.

(i) Specific heat at constant volume (C_v)

(ii) Specific heat at constant pressure (C_p)

If we consider a unit mass of gas, these specific heats are called the principal specific heats of the gas. In this case, c_v denotes the specific heat of the gas at constant volume. It is defined as the amount of heat required to raise the temperature of 1 gram of the gas through 1⁰C at constant volume. The specific heat of the gas at constant pressure is denoted by c_p. It is defined as the amount of heat required to raise the temperature of 1 gram of the gas through 1⁰C at constant pressure.

Instead of considering one gram of the gas, if we consider one mole of the gas, then specific heats are called gram molecular or molar specific heats of the gas.

The molar specific heat of a gas at constant volume is defined as the amount of heat required to raise the temperature of 1 gram mole of gas through 1 K (or 1 ⁰C) at constant volume. It is denoted by C_v. If M is the molecular weight of the gas in gram, then

Let an amount of heat dQ be supplied to one gram mole of a gas at constant volume. Let dT be the increase in temperature.

The molar specific heat of a gas at constant pressure is defined as the amount of heat required to raise the temperature of 1 gram mole of gas through 1 K (or 1 ⁰C) at constant pressure. It is denoted by C_p. If M is the molecular weight of the gas in gram, then

Let an amount of heat dQ be supplied to one gram mole of gas at constant pressure. Let dT be the increase in temperature.

Both C_p and C_v are measured in J K^-1 mol^-1.

C_p is greater than C_v. If a gas is heated at constant volume, the gas does no work against external pressure. In this case, the whole of the heat energy supplied to the gas is spent in raising the temperature of the gas.

If a gas is heated at constant pressure, its volume increases. In this case, heat energy is required for the following two purposes:

(i) To increase the volume of the gas against external pressure.

(ii) To increase the temperature of 1 mole of gas through 1 K.

Thus, more heat energy is required to raise the temperature of 1 gram mole of gas through 1 K when it is heated at constant pressure than when it is heated at constant volume.The difference between Cp and C_v is the thermal equivalent of the work done by the gas in expanding against external pressure.