Specific heat capacity of an ideal gas
- We have defined specific heat capacity and molar specific heat capacity earlier in the previous chapter.
- There are two specific heats of ideal gases.
(i) Specific heat capacity at constant volume
(ii) Specific heat capacity at constant pressure
C_{p} and C_{v} are molar specific heat capacities of ideal gas at constant pressure and volume respectively for C_{p} and C_{v} of ideal gas there is a simple relation.
$C_p-C_v = R$ ---(1)
where R- universal gas constant
- This relation can be proved as follows from first law of thermodynamics for 1 mole of gas we have
$ \Delta Q =\Delta U+P \Delta V $ --(2)
- If heat is absorbed at constant volume for the temperature difference $\Delta T$ then ΔV = 0 and
C_{V}=(ΔQ/ΔT)_{V}=(ΔU/ΔT)_{V} -----(3)
If Q in absorbed at constant pressure for the temperature difference $\Delta T$,then
C_{P}=(ΔQ/ΔT)_{P}=(ΔU/ΔT)_{P}+P(ΔV/ΔT)_{P} ---- (4)
Now ideal gas equation for 1 mole of gas is
PV = RT
P(ΔV/ΔT) = R --(5)
From (3) and (4)
C_{P} - C_{V}=(ΔU/ΔT)_{P}-(ΔU/ΔT)_{V}+P(ΔV/ΔT)_{P}
Now from equation (5)
C_{P} - C_{V}=(ΔU/ΔT)_{P}-(ΔU/ΔT)_{V} + R
- Since internal energy U of ideal gas depends only on temperature so subscripts P and V have no meaning.
$C_P - C_V = R$
which is the desired relation
Relation Between C_{V} and Internal Energy
From equation (3)
C
_{V}=(ΔQ/ΔT)
_{V}=(ΔU/ΔT)
_{V}
or
$\Delta U = C_v \Delta T$
or
$dU= C_v dT$
for n moles for gas
$dU= nC_v dT$
If we take the internal energy at T=0,then
$U=nC_v T$
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