$C_{P} - C_{V}$ for an ideal gas is $R$ .
State whether the statement is True or False

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Solution

According to the first law of thermodynamics:
$Δ Q = Δ U + Δ W \Rightarrow Δ W = P Δ V$
$Δ Q_{P} = Δ U + P Δ V$

Since $Δ Q_{p} = n C_{p} Δ T$
and $Δ U = n C_{v} Δ T$ , therefore,
$n C_{p} Δ T = n C_{v} Δ T + P Δ V (1)$

We know that $P V = n R T$
At $T_{1} K e l v i n : P V_{1} = n R T_{1}$ .....(a)
At $T_{2} K e l v i n : P V_{2} = n R T_{2}$ .....(b)

Subtracting (a) from (b)
$P V_{2} - P V_{1} = n R T_{2} - n R T_{1}$
$P (V_{2} - V_{1}) = n R (T_{2} - T_{1})$
$P Δ V = n R Δ T$

Putting the value of $P Δ V$ in equation (1)
$n C_{p} Δ T = n C_{v} Δ T + n R Δ T$
$n C_{p} Δ T = n Δ T (C_{v} + R)$
$C p = (C v + R)$
$C p - C v = R$
Hence the statement is True

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