To determine the heat capacity of the system you've described, we need to analyze the behavior of the ideal monoatomic gas in the thermally insulated vessel. Since the vessel is insulated, there is no heat exchange with the environment, and we can focus on the internal energy changes and the work done by the gas as it expands against the spring. Let's break this down step by step.
Understanding the System
We have a thermally insulated vessel divided into two parts by a frictionless piston. On the left side, there is one mole of an ideal monoatomic gas, while the right side is initially empty. The piston is connected to a spring, which means that as the gas expands, it will do work on the spring.
Key Concepts
- Ideal Gas Behavior: For an ideal monoatomic gas, the internal energy (U) is given by the equation U = (3/2)nRT, where n is the number of moles, R is the universal gas constant, and T is the temperature in Kelvin.
- Work Done by the Gas: When the gas expands, it does work on the piston and the spring. The work done (W) can be expressed as W = ∫PdV, where P is the pressure and dV is the change in volume.
- Heat Capacity: The heat capacity (C) of a system is defined as the amount of heat required to change the temperature of the system by one degree Celsius (or one Kelvin).
Calculating the Heat Capacity
In this scenario, since the vessel is insulated, the first law of thermodynamics applies, which states that the change in internal energy (ΔU) is equal to the work done (W) on the system plus the heat added (Q). However, since the system is insulated, Q = 0. Therefore, we have:
ΔU = -W
For one mole of an ideal monoatomic gas, the change in internal energy when the temperature changes by ΔT is:
ΔU = (3/2)nRΔT = (3/2)(1)(R)ΔT = (3/2)RΔT
Now, let's consider the work done by the gas as it expands against the spring. The work done can be expressed in terms of the spring constant (k) and the displacement (x) of the spring:
W = (1/2)kx²
Since the gas expands and does work on the spring, we can relate the work done to the change in internal energy:
(3/2)RΔT = - (1/2)kx²
Finding the Heat Capacity
To find the heat capacity of the system, we need to express it in terms of the change in temperature. The heat capacity at constant volume (C_v) for one mole of a monoatomic ideal gas is:
C_v = (3/2)R
However, in this case, we need to consider the work done against the spring as well. The effective heat capacity (C_eff) of the system can be derived from the relationship between the change in internal energy and the work done:
C_eff = ΔU / ΔT = -W / ΔT
Substituting the expressions we have:
C_eff = - (1/2)kx² / ΔT
To find a specific value for the heat capacity, we would need to know the spring constant (k) and the displacement (x) of the spring during the expansion. However, the general form of the heat capacity for this system, considering the work done against the spring, can be expressed as:
C_eff = (3/2)R + (1/2)(k/x)Δx
In summary, the heat capacity of the system depends on both the properties of the gas and the characteristics of the spring. The effective heat capacity will be greater than that of the gas alone due to the work done on the spring during the expansion process.
