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Advanced Physics Topics like quantum mechanics and relativity have revolutionized our understanding of the universe.
What is Specific Heat at Constant Pressure Called?
From calorimetry (see the chapter Calorimetry), we know that H = Inst or, s = \(\frac{H}{m t}\)
i.e., specific heat \(=\frac{\text { heat transfer }}{\text { mass } \times \text { change in temperature }}\)
For a body of unit mass, specific heat (s)
\(=\frac{\text { heat transfer }(H)}{\text { change in temperature }(t)}\)
This definition of specific heat is useful for solids and liquids, but is incomplete in case of gases. For example, if a gas is suddenly compressed, its temperature rises, even when no heat is exchanged with the surroundings. Here, H = 0, but t ≠ 0.
So, s = \(\frac{H}{t}\) = \(\frac{0}{t}\) = o.
Again, if the temperature of an expanding gas is to be kept constant, some heat must be supplied from outside. Here,
H ≠ 0, but t = 0.
So, s = \(\frac{H}{t}\)
i.e., when change in temperature = 0, specific heat → ∞
This shows that the specific heat of a gas can have any value from zero to infinity. So, the specific heat can have a definite and useful value only when some condition is imposed on the gas.
In general, when a gas is heated, not only does its temperature change, but the volume and the pressure also change at the same time. Now, two special cases are considered:
- The absorbed heat increases the temperature and pressure of the gas while the volume is kept constant.
- The absorbed heat increases the temperature and volume of the gas while the pressure is kept constant.
Heat required by unit mass of a gas to raise its temperature by one degree, keeping its volume or pressure constant, is called specific heat of the gas. So, we get definitions of two specific heats in case of a gas-
- specific heat at constant volume and
- specific heat at constant pressure.
Specific heat at constant volume: Heat absorbed by unit mass of a gas to raise its temperature by one degree, keeping the volume constant, is called the specific heat of that gas at Constant volume (cv).
So, heat taken by a gas of mass m for rise in temperature t, at constant volume, is
Qv = mcvt ………. (1)
Specific heat at constant pressure: Heat absorbed by unit mass of a gas to raise its temperature by one degree, keeping the pressure constant, is called the specific heat of that gas at constant pressure (cp).
So, heat taken by a gas of mass rn for rise in temperature t, at constant pressure, is
Qp = mcpt ……….. (2)
Heat capacities: Heat capacity of a body is defined as the heat absorbed by the body per unit rise in temperature. The above discussions show that appropriate conditions are to be imposed on the definition of heat capacity also.
So, heat capacity at constant volume,
Cv \(=\frac{\text { heat absorbed }}{\text { change in temperature }}\) = \(\frac{Q_\nu}{t}\) = mcv [using (1)]
or, heat capacity at constant volume = mass × specific heat at constant volume
Similarly, Cp = mcp
or, heat capacity at constant pressure = mass × specific heat at constant pressure
Specific heat is an intrinsic thermodynamic property. But the relations Cv = mcv, and Cp = mcp, show that Cv and Cp are proportional to the mass. So, heat capacity is an extrinsic thermodynamic property.
Molar heat capacity or molar specific heat: If M is the molecular weight of a gas (or of any other substance), mass of 1 mol = M g. The heat capacity of 1 mol of a substance is called molar heat capacity or molar specific heat.
Molar specific heat at constant volume, Cv = Mcv
and molar specific heat at constant pressure, Cp = Mcp
The molecular weight M is constant for a particular substance. So, like cv, and cp, the molar specific heats Cv and Cp are also intrinsic thermodynamic properties.
Units of specific heat
Specific heat of solids and liquids: when a solid or a liquid is heated, thermal expansion of volume takes place. But, a change in pressure due to a change in temperature is not usually observed in practice. So, for solids and liquids, the specific heat at constant volume (cv) is not a useful property. The specific heat of a solid or a liquid, as discussed in calorimetry, is actually its specific heat at constant pressure (cp). The observation of cv may be very difficult, but still every solid and liquid has definite values of both cv and cp, just like that of a gas.
For water, the specific heat at constant pressure is
cp = 1 cal ᐧ g-1°C-1 (CGS)
= 4200 J ᐧ kg-1 ᐧ K-1 (SI)
As M = 18 for water, the molar specific heat at constant pressure is,
Cp = 18 cal ᐧ mol-1 ᐧ °C-1 (CGS)
= 18 × 4.2 J ᐧ mol-1 ᐧ K-1 (SI)
Relation with internal energy:
The first law of thermodynamics states that
dQ = dU + dW or, dQ = dU + pdV ……. (3)
If we take a process at constant volume, the heat absorbed by a mass m for a temperature change dT may be expressed as dQv = mcvdT. Here, as V = constant, dv = 0. So from equation (3), we get,
mcvdT = dU or, dU = mcvdT ……… (4)
When m, cv and dT arc known, equation (4) gives a measure of the change of internal energy In integral form, Uf – Ui = mcv(Tf – Ti). It is also clear that dU or Uf – Ui is proportional to mass. So, the difference in internal energy, and the internal energy itself, is an extrinsic thermodynamic property.
cp is greater than cv:
The first law of thermodynamics gives,
dQ = dU + dW = dU + pdV
For a process at constant volume, dQv = mcvdT and dV = 0. So, mcvdT = dU or, dU = mcvdT.
Now let us consider an ideal gas as the working substance. For a process at constant pressure with the same temperature change dT, we have dQp = mcpdT. As the internal energy U of a gas depends on temperature only, the value of dU in this process will be the same, i.e., dU = mcvdT. So from the relation dQ = dU+ dW, we have for the process at constant pressure,
dQp = mcvdT + dW
or, mcpdT = mcvdT + pdV or, m(cp – cv)dT = pdV
or, cp – cv = \(\frac{p}{m} \frac{d V}{d T}\) ………. (5)
As volume of gas increses with temperature rise, \(\frac{p}{m} \frac{d V}{d T}\) > 0.
∴ Equation (5) shows that cp > cv for a gas.
In general, this is true for solids and liquids also. For any substance, the specific heat at constant pressure is greater than the specific heat at constant volume. This is because at constant volume, no work is done. So the heat absorbed changes the internal energy only. But at constant pressure, the heat absorbed changes the internal energy and also does some work. Thus, a greater amount of heat is absorbed in the latter case, and as a result, the specific heat at constant pressure is higher.
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