Question

When the gas is heated at constant volume, the heat supplied increases the internal energy of the gas: (A) True (B) False

Answer 1

Hint: The result can be interpreted through Maxwell equation and the combination of first and second law of thermodynamics.

Complete step by step solution:
Second law of thermodynamics states that
“There exists for every thermodynamic system in equilibrium an extensive scalar property called the entropy, S, such that in an infinitesimal reversible change of state of the system, dS = dQ/T, where T is the absolute temperature and dQ is the amount of heat received by the system. The entropy of a thermally insulated system cannot decrease and is constant if and only if all processes are reversible.”
In simple terms, we can define the second law of thermodynamics that when energy changes from one form to another form, or matter moves freely, entropy (disorder) in a closed system increases. Differences in temperature, pressure and density tend to even out horizontally after a while. This law is very important because it talks about entropy and tells whether or not a process is going to be spontaneous or not. It provides the criterion for the feasibility of any process. A process cannot occur unless it satisfies both the first and second law of thermodynamics.
Entropy is a measure of the randomness of the system or it is the measure of energy or chaos within an isolated system. It can be considered as a quantitative index that describes the quality of energy.
Meanwhile, there are few factors that cause an increase in entropy of the closed system. Firstly, in a closed system, while the mass remains constant there is an exchange of heat with the surroundings. This change in the heat content creates a disturbance in the system thereby increasing the entropy of the system.
Secondly, internal changes may occur in the movements of the molecules of the system. This leads to disturbances which further causes irreversibilities inside the system resulting in the increment of its entropy.

Mathematically, it can be expressed as
First law of thermodynamics, mathematically, represented as
               \[\text{dU=dQ-dW}\]\[\to \text{1}\]
Work and Heat exchange in terms of stable variables:
      \[\text{dQ=TdS}\]
     \[\text{dW=PdV}\]\[\to \text{ Only true for reversible processes (2nd law of thermodynamics)}\]
Putting the equation in 1, we have
               \[\text{dU=dQ-dW}\]
               = \[\text{TdS-PdV}\]
It can also be derived from Maxwell equations.
Therefore, the heat supplied increases the internal energy of the gas and increases the temperature of the gas.
Hence, the given option is (A) True.

Note: The above equation can also be derived by Maxwell equations. There is also a trick called Thermodynamic Square through which it can easily be derived.
Mathematically, the second law of thermodynamics is represented as-
        \[\Delta {{\text{S}}_{\text{univ}}}>0\]
where $\Delta {{\text{S}}_{\text{univ}}}$ is the change in the entropy of the universe.