Question
Define molar specific heat capacities at constant volume and pressure. Considering thermodynamical process in a cylinder with parameters P, V and T, derive the Mayer's relation.
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Answer
Molar specific heat capacity is the heat energy required to raise the temperature of 1 mole of a substance by 1K and expressed in J mol-1 K-1.
$\text{C}=\frac{\text{Q}}{1\text{mole 1}\text{K}}$
Depending on the condition that whether volume or pressure is constant, molar specific heat is written as Cv and Cp.
Relation between Cp and Cv: Suppose one mole of a gas is heated so that its temperature rises by dT.
Heat supplied $=1\times\text{C}_\text{V}\times\text{C}_\text{V}\text{dT}\dots(\text{i})$
Since the volume is constant, the gas will not perform external work in accordance with the first law of thermodynamics and the heat supplied will be just equal to the increase in the internal energy of the gas.
$\therefore\text{dU}=\text{C}_{\text{V}}\text{dT}\dots\text{(ii)}$
Let the gas be heated at constant pressure to again increase its temperature by dT, and dQ be the amount of heat supplied, therefore,
dQ = 1 × CP × dT = CPdT ...(iii)
The heat supplied at a constant pressure increases the temperature by dT hence increases its internal energy by dU = CvdT as well as enables the gas to perform work dW.
dW = PdV ...(iv)
From the first law of thermodynamics, we have
dQ = dU + dW
Substituting the values, we get,
$\text{C}_\text{P}\text{dT}=\text{C}_\text{v}\text{dT}+\text{PdV}$
But PV = RT (For one mole of the gas)
or PdV = RdT
$\therefore\text{C}_\text{P}\text{dT}=\text{C}_\text{V}\text{dT}$
$\text{C}_\text{P}-\text{C}_\text{V}=\text{R}$
This is the relation between two principal specific heats of the gas when Cp, Cv and R are measured in the units of either heat or of work.
$\text{C}=\frac{\text{Q}}{1\text{mole 1}\text{K}}$
Depending on the condition that whether volume or pressure is constant, molar specific heat is written as Cv and Cp.
Relation between Cp and Cv: Suppose one mole of a gas is heated so that its temperature rises by dT.
Heat supplied $=1\times\text{C}_\text{V}\times\text{C}_\text{V}\text{dT}\dots(\text{i})$
Since the volume is constant, the gas will not perform external work in accordance with the first law of thermodynamics and the heat supplied will be just equal to the increase in the internal energy of the gas.
$\therefore\text{dU}=\text{C}_{\text{V}}\text{dT}\dots\text{(ii)}$
Let the gas be heated at constant pressure to again increase its temperature by dT, and dQ be the amount of heat supplied, therefore,
dQ = 1 × CP × dT = CPdT ...(iii)
The heat supplied at a constant pressure increases the temperature by dT hence increases its internal energy by dU = CvdT as well as enables the gas to perform work dW.
dW = PdV ...(iv)
From the first law of thermodynamics, we have
dQ = dU + dW
Substituting the values, we get,
$\text{C}_\text{P}\text{dT}=\text{C}_\text{v}\text{dT}+\text{PdV}$
But PV = RT (For one mole of the gas)
or PdV = RdT
$\therefore\text{C}_\text{P}\text{dT}=\text{C}_\text{V}\text{dT}$
$\text{C}_\text{P}-\text{C}_\text{V}=\text{R}$
This is the relation between two principal specific heats of the gas when Cp, Cv and R are measured in the units of either heat or of work.
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