My teacher told that at constant temperature $C_m$ tends to infinity but $C_v$ and $C_p$ do not.

The context was calculating ΔH and ΔU in an isothermal reversible process for ideal gases. The explanation he gave was:

$$\Delta U=nC_v\Delta T=0$$

(since $\Delta T=0$ for isothermal process) and:

$$\Delta H=nC_p\Delta T=0$$

(since $\Delta T=0$ for isothermal process). Therefore:

$$\Delta U=q+w=0$$

or:

$$q=-w=nC_m\Delta T$$

Here too, $\Delta T=0$ but $q=-w\ne 0$ (since $C_m\to\infty$). Therefore we calculate $w$ by:

$$w=\int P_{ext}dV$$

Now I wish to know the explanation of this different behavior of different heat capacities at a constant temperature. (All symbols have their standard meaning.)

My teacher told that at constant temperature $C_m$ tends to infinity but $C_v$ and $C_p$ do not.

The context was calculating ΔH and ΔU in an isothermal reversible process for ideal gases. The explanation he gave was:

$$\Delta U=nC_v\Delta T=0$$

(since $\Delta T=0$ for isothermal process) and:

$$\Delta H=nC_p\Delta T=0$$

(since $\Delta T=0$ for isothermal process). Therefore:

$$\Delta U=q+w=0$$

or:

$$q=-w=nC_m\Delta T$$

Here too, $\Delta T=0$ but $q=-w\ne 0$ (since $C_m\to\infty$). Therefore we calculate $w$ by:

$$w=\int P_{ext}dV$$

He gave the reason of molar heat capacity to be ∞ as the heat required to raise the temperature of one mole of the substance by 1K at a constant temperature. (Well this sounds funny.) Now I wish to know the explanation of this different behavior of different heat capacities at a constant temperature. (All symbols have their standard meaning. Cm means molar heat capacity)

My teacher told that at constant temperature, Cm tends to infinity but Cv or Cp does not. The context was calculating ΔH and ΔU in an isothermal reversible process for ideal gases. The explanation he gave was ΔU=nCvΔT=0 (since ΔT=0 for isothermal process) and ΔH=nCpΔT=0 (since ΔT=0 for isothermal process) Therefore, ΔU=q+w=0 or, q=-w=nCmΔT. Here too, ΔT=0 but q=-w≠0 (since Cm→∞). Therefore we calculate w by w=∫PextdV Now I wish to know the explanation of this different behavior of different heat capacities at a constant temperature. (All symbols have their standard meaning)

My teacher told that at constant temperature $C_m$ tends to infinity but $C_v$ and $C_p$ do not. 

The context was calculating ΔH and ΔU in an isothermal reversible process for ideal gases. The explanation he gave was:

$$\Delta U=nC_v\Delta T=0$$

(since $\Delta T=0$ for isothermal process) and:

$$\Delta H=nC_p\Delta T=0$$

(since $\Delta T=0$ for isothermal process). Therefore:

$$\Delta U=q+w=0$$

or:

$$q=-w=nC_m\Delta T$$

Here too, $\Delta T=0$ but $q=-w\ne 0$ (since $C_m\to\infty$). Therefore we calculate $w$ by:

$$w=\int P_{ext}dV$$

Now I wish to know the explanation of this different behavior of different heat capacities at a constant temperature. (All symbols have their standard meaning.)