Isothermal Process : Definition, Conditions and Equations

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What is the Isothermal Process on the P-V Diagram?

Isothermal process: A process in which the temperature of a system remains constant is called an isothermal process.

The changes in volume, pressure and other quantities in isothermal processes are called isothermal changes.

Let a gas be enclosed inside a metallic cylinder-piston arrangement. The cylinder and the piston are made of a conducting material. The piston can move without friction along the inner walls of the cylinder. Now, if the gas expands very slowly, an external work is done by the gas. As a result, the internal energy of the gas will tend to fall, as it supplies the energy necessary to do the work.

But the internal energy of a gas depends on its temperature. So the temperature of the gas will also tend to fall. As metal is a good conductor of heat and the expansion of the gas is very slow, heat will enter the metallic cylinder from its surroundings and keep the temperarure of the gas constant. This is called an Isothermal expansion because there is no change in temperature during expansion.

Similarly, if the gas contracts very slowly, work is done on the gas. For that, heat is evolved and temperature of the gas increases. As the compression of the gas is very slow heat evolved will be transmitted to the surroundings through the conducting cylinder, so that the temperature of the gas remains constant. Such type of compression of a gas is called an Isothermal compression.

For an ideal gas, if the temperature remains constant, the internal energy also remains constant.
So, U_f = U_i or, U_f – U_i = 0
Then from the first law of thermodynamics,
Q = (U_f – U_i) + W = 0 + W or, Q = W
i.e., heat taken from the surroundings = external work done.

For an isothermal compression, both Q and W are negative. This means that work is done on the system by its surroundings and the system loses an equal amount of heat.

Isothermal conditions:

i) The heat exchange between the system and its surroundings should occur very fast. So the gas should be kept in a good conducting container and surrounded by a medium of high thermal capacity.

ii) The process should be very slow, so that the time is sufficient for heat exchange to keep the temperature constant. Because this expansion or compression of a gas should take place very slowly, an isothermal process is a very slow process and any slow thermal process is usually regarded as isothermal.

Isothermal process of an ideal gas on a PV-diagram: For n mol of an ideal gas, pV = nRT. For an isothermal process, T = constant. So, pV = constant

This equation is represented by a rectangular hyperbola on a pV diagram [Fig.]. Isothermal process at higher temperatures have higher values of RT, i.e., higher values of pV. So the corresponding curves will be at greater distances from the origin.

The curves shown in Fig. are called isothermal curves for an ideal gas. No two curves can intersect, because a point of intersection denotes two values of temperature of a gas simultaneously. This can never happen.

Work done by an ideal gas in an isothermal process: For change in volume dV at constant temperature of a certain amount of an ideal gas, work done dW = pdV. If the volume changes from V_i to V_f, the total work done is
W = \(\int_{V_i}^{V_f} p d V\)
= the area ABCDA on the pV-diagram [Fig.]

For n mol of an ideal gas, pV = nRT or, p = \(\frac{n R T}{V}\)
For an isothermal process, T = constant.
∴ W = \(\int_{V_i}^{V_f} \frac{n R T}{V} d V\) = nRT\(\int_{V_i}^{V_f} \frac{d V}{V}\)
= nRT ln\(\frac{V_f}{V_i}\) [lnx = log_ex]
Again, from Boyle’s law for an isothermal process,
p_iV_i = p_fV_f or, \(\frac{V_f}{V_i}\) = \(\frac{p_i}{p_f}\)
So, W = nRT ln\(\frac{V_f}{V_i}\) = nRT ln\(\frac{p_i}{p_f}\) ……. (1)
For isothermal expansion, V_f > V_i, so the ratio V_f/V_f is greater than 1.
Hence, W = nRTln\(\frac{V_f}{V_i}\) > o
Now for an isothermal compression of a gas. V_f < V_i. Hence, W = nRTln\(\frac{V_f}{V_i}\) < 0.
So for an isothermal expansion work done is positive while for an isothermal compression, associated work done is negative.

For an ideal gas, U = constant, when T = constant. So from the first law of thermodynamics,
Q = (U_f – U_i) + W = 0 + W = W.
Thus from equation (1),
Q = nRT ln\(\frac{V_f}{V_i}\) = nRTln\(\frac{p_i}{p_f}\)…….. (2)