Question

Three gaseous systems, \(G_1, G_2\), and \(G_3\) with pressure and volume (\(P_1, V_1\)), (\(P_2, V_2\)), and (\(P_3, V_3\)), respectively, are such that
(I) when \(G_1\) and \(G_2\) are in thermal equilibrium, \(P_1V_1 - P_2V_2 + \alpha P_2 = 0\), is satisfied, and
(II) when \(G_1\) and \(G_3\) are in thermal equilibrium, \(P_3V_3 - P_1V_1 + \frac{\beta P_1V_1}{V_3} = 0\), is satisfied.
The relation(s) valid at thermal equilibrium is(are)
(\(\alpha\) and \(\beta\) are constants of appropriate dimensions)

• A. \(P_3V_3 - (P_2V_2 - \alpha P_2) \left(1 - \frac{\beta}{V_3}\right) = 0\)
• B. \(P_3V_3 + (P_2V_2 + \alpha P_2) \left(1 + \frac{\beta}{V_3}\right) = 0\)
• C. \(P_1V_1 = P_2V_2 = P_3V_3\)
• D. \(P_3V_3 + P_1V_1 \left(\frac{\beta}{V_3} - 1\right) = 0\)

Hint:

The Zeroth Law implies the existence of a state function, temperature. Any quantity that is equal for two systems in thermal equilibrium can be taken as a measure of temperature. In this problem, \(P_1V_1\) acts as the "thermometer." You can express the temperatures of \(G_2\) and \(G_3\) in terms of their own variables, and then equate them to find the relationship when they are in equilibrium with each other.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem is based on the Zeroth Law of Thermodynamics. The law states that if two systems are each in thermal equilibrium with a third system, then they are in thermal equilibrium with each other. This implies that there exists a property, which we call temperature, that is the same for all systems in thermal equilibrium. Any function of the state variables (like P and V) that is equal for two systems in thermal equilibrium can be used to define an empirical temperature scale.
Step 2: Key Formula or Approach:
From (I): \(P_1V_1 = P_2V_2 - \alpha P_2\)
From (II): \(P_3V_3 - P_1V_1 + \frac{\beta P_1V_1}{V_3} = 0 \implies P_3V_3 = P_1V_1 \left(1 - \frac{\beta}{V_3}\right)\)
Now, we can substitute the expression for \(P_1V_1\) from (I) into (II).
\[ P_3V_3 = (P_2V_2 - \alpha P_2) \left(1 - \frac{\beta}{V_3}\right) \] Rearranging this gives:
\[ P_3V_3 - (P_2V_2 - \alpha P_2) \left(1 - \frac{\beta}{V_3}\right) = 0 \] This exactly matches option (A).
Let's check option (D).
From (II), we have \(P_3V_3 - P_1V_1 + \frac{\beta P_1V_1}{V_3} = 0\). Factor out \(P_1V_1\):
\[ P_3V_3 - P_1V_1 \left(1 - \frac{\beta}{V_3}\right) = 0 \] This can be rewritten as: \[ P_3V_3 + P_1V_1 \left(-1 + \frac{\beta}{V_3}\right) = 0 \implies P_3V_3 + P_1V_1 \left(\frac{\beta}{V_3} - 1\right) = 0 \] This exactly matches option (D). Let's check other options.
(B) has plus signs that don't match our derivation for (A).
(C) implies \(\alpha=0\) and \(\beta=0\), which is not generally true.
So, (A) is a relation between \(G_2\) and \(G_3\) when they are in thermal equilibrium. (D) is just a restatement of the given condition (II) for equilibrium between \(G_1\) and \(G_3\). Since the question asks for relation(s) valid at thermal equilibrium, and (D) is given as one such relation, it is a valid choice.
Step 4: Final Answer:
By substituting the condition for equilibrium between \(G_1\) and \(G_2\) into the condition for equilibrium between \(G_1\) and \(G_3\), we derive a condition for equilibrium between \(G_2\) and \(G_3\), which is option (A). Option (D) is just a simple algebraic rearrangement of the given condition (II). Both are mathematically valid relations that hold when the systems are in thermal equilibrium. Thus, options (A) and (D) are correct.