I have trouble to understand that if

$$\frac{\mathrm{d}q}{\mathrm{d}t} = n\cdot C $$

then during an isothermal process change in heat must also be zero , but why do we say only change in internal energy equals to zero?

  • $\begingroup$ what is n in your equation? $\endgroup$ – Bob D Nov 7 '18 at 12:46

In thermodynamics, the heat capacity is regarded as the derivative of the internal energy with respect to temperature (a function of state) rather than the derivative of the heat with respect to temperature (a function of process path). This is because we regard heat capacity as a physical property of the material. The derivative of the heat with respect to temperature only gives the correct answer if no work is done.


Heat can transfer to or from a substance without causing a temperature and internal energy change of the substance. And a temperature and internal energy change can occur to a substance without heat transfer to or from the substance.

To understand the reasons for this you need to keep in mind that there are two possible forms of energy transfer: Heat (Q) and work (W). The two forms are related to the internal energy of a substance by the first law, which can be written, in the case of a closed system (no mass transfer), as:

$$\Delta U=Q-W$$

Where $Q$ is positive if heat is added to the system (energy in) and $W$ is positive if work is done by the system (energy out).

In the case of an isothermal process, the heat into or out of the system is exactly equal to the work done by or on the system, respectively, so that the change in internal energy, and thus temperature, is zero. This demonstrates, as pointed out by @Chester Miller, that your equation only applies in the absence of work.

There is another kind of isothermal process where your equation does not apply even in the absence of work. For this we have for $W=0$ and:

$$\Delta U=Q$$

In this case heat transfer is occurring to (or from) a substance undergoing a phase change at constant temperature. An example is melting ice at 1 atm and 0 C. In this example we have:

$$\Delta U=Q=mh$$

Where $h$ is the latent heat of fusion of the substance.

Now how can the temperature and internal energy change without heat transfer ($Q$)? From the first law if $Q=0$, then:

$$\Delta U=-W$$

So if work is done on or by the system it can cause a change in internal energy and temperature. An example is the compression a gas in an insulated cylinder. The temperature of the gas will increase, and yet no heat transfer has occurred.

Hope this helps.

  • $\begingroup$ Note that the sign of $W$ is a matter of convention; in many places, $W$ is work recieved by the system (just as heat), in which case the 1st law reads $\Delta U=Q+W$ $\endgroup$ – Nicolas Nov 7 '18 at 15:23
  • $\begingroup$ @Nicolas Yes that is often how it is expressed in chemistry. But when it is, W is positive if done one the system and negative if done by the system (reverse of the case where delta U = Q-W). It doesn't matter since in both cases energy in is a positive change in internal energy and energy out a negative change, $\endgroup$ – Bob D Nov 7 '18 at 17:09
  • $\begingroup$ Indeed, I was merely pointing that out in case someone wonders why the sign is opposite in many textbooks. $\endgroup$ – Nicolas Nov 8 '18 at 13:08

Internal Energy $U$ is related to the temperature $T$ by

$U = cmT$

with heat capacity $c$ and the mass $m$. However, heat is defined by the first law of thermodynamics as

$dq = dU-dW$

where $W$ is mechanical, chemical or other form of work. Heat is defined as the Energy difference from processes that are irreversible. In an isothermal process, there is $dU=0$ (see above equation) and therefore

$dq = -dW$

  • $\begingroup$ Internal energy is not only related to temperature. It is the sum of all the kinetic and potential molecular energies. You can have a change in internal energy without temperature change (e.g. constant temperature phase changes). $\endgroup$ – Bob D Nov 7 '18 at 12:49
  • $\begingroup$ Clear, in general, internal Energy can Change without temperature Change, but frequently the equation given above is used. $\endgroup$ – kryomaxim Nov 7 '18 at 13:57
  • $\begingroup$ "Heat is defined as the Energy difference from processes that are irreversible" I think that's not quite true, you can have heat exchange in a reversible process (e.g. isothermal) $\endgroup$ – Nicolas Nov 7 '18 at 15:16
  • 1
    $\begingroup$ @kryomaxim $U=cmT$ only holds for an ideal gas. Actually, that's often used as a definition of an ideal gas… It's frequently given as an example, as ideal gas low is as simple as equations of state can get. $\endgroup$ – Nicolas Nov 7 '18 at 15:17
  • $\begingroup$ Also, incompressible liquids and solids have this caloric equation of state. $\endgroup$ – kryomaxim Nov 7 '18 at 16:07

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