Thermodynamics

Step 1: Understanding the Concept:
The TdS equations relate the change in entropy (S) to other state variables like temperature (T), volume (V), and pressure (P). They are derived from the first law of thermodynamics combined with Maxwell's relations. The "first" TdS equation expresses dS in terms of dT and dV.
Step 2: Key Formula or Approach:
We start by considering entropy S as a function of temperature T and volume V, i.e., . The total differential is:
We need to evaluate the two partial derivatives.
Step 3: Detailed Explanation:
1. Evaluate :
From the definition of heat capacity at constant volume, .
Rearranging this gives: .
2. Evaluate :
To evaluate this term, we use a Maxwell relation. The Maxwell relation derived from the Helmholtz free energy ( ) is:
3. Substitute back into the dS equation:
Substitute the expressions for the partial derivatives back into the total differential for dS:
4. Obtain the TdS form:
Multiply the entire equation by T to get the first TdS equation:
Step 4: Final Answer:
The correct form of the first TdS equation is . This matches the expression .

Step 1: Understanding the Concept:
This question requires matching fundamental equations and definitions from thermodynamics with their respective names or descriptions.
Step 2: Detailed Explanation:

(A) Clausius-Clapeyron equation: This equation relates the slope of the coexistence curve between two phases of matter on a pressure-temperature (P-T) diagram to the latent heat (L) and the change in volume (V₂ - V₁) during the phase transition. The equation is . This matches with (IV).

(B) Gibbs Function (G): The Gibbs free energy is a thermodynamic potential defined as . Since Enthalpy , the Gibbs function can be written as . This matches with (III).

(C) Enthalpy (H): Enthalpy is a thermodynamic potential defined as the sum of the internal energy (U) and the product of pressure and volume (PV). The equation is . This matches with (II).

(D) Adiabatic change in Perfect Gas: An adiabatic process is one that occurs without heat or mass transfer. For a perfect gas undergoing a reversible adiabatic process, the relationship between pressure (P) and volume (V) is given by , where is the heat capacity ratio. This matches with (I).
Step 3: Final Answer:
Based on the analysis, the correct pairings are:
(A) - (IV)
(B) - (III)
(C) - (II)
(D) - (I)
This corresponds to option (A).

Step 1: Understanding the Concept:
Entropy is a measure of the disorder or randomness of a system. The change in entropy ( ) for a reversible process is defined as . We need to find the expression for the total entropy change for an isochoric (constant volume) process involving a perfect gas.
Step 2: Key Formula or Approach:
From the first law of thermodynamics, the heat added to a system is .
For a perfect gas, the change in internal energy is and the work done is .
Therefore, for a reversible process:
Dividing by T, we get the general expression for entropy change:
We will apply the condition for an isochoric process to this equation.
Step 3: Detailed Explanation:
An isochoric process is one that occurs at constant volume.
This means , and the change in volume .
Substituting into the general entropy equation:
To find the total change in entropy from state 1 to state 2, we integrate this expression:
For a perfect gas at constant volume, the pressure and temperature are related by Gay-Lussac's Law:
Substituting this into our entropy equation (assuming 1 mole, or that represents the total heat capacity):
Step 4: Final Answer:
The change of entropy for a perfect gas during an isochoric process is .

Step 1: Understanding the Concept:
The question asks for the ratio of two coefficients of thermal expansion: the adiabatic coefficient and the isobaric coefficient. The coefficient of volume expansion describes how the volume of a substance changes with a change in temperature.
Step 2: Key Formula or Approach:
The isobaric (constant pressure) coefficient of volume expansion is defined as:
The adiabatic (constant entropy) coefficient of volume expansion is defined as:
We need to find the ratio for a perfect gas.
Step 3: Detailed Explanation:
1. Calculate the isobaric coefficient ( ):
For one mole of a perfect gas, . At constant pressure, .
So, .
2. Calculate the adiabatic coefficient ( ):
For an adiabatic process, . Differentiating this with respect to T:
Rearranging to find for this process:
So, .
3. Find the ratio :
This can be rewritten as:
Step 4: Final Answer:
The ratio of the adiabatic to the isobaric coefficient of expansion is .

Step 1: Understanding the Concept:
The work done on a gas during an adiabatic compression is given by the change in its internal energy. We can calculate this work using the initial and final states (pressure and volume) of the gas. The process is adiabatic, so there is no heat exchange with the surroundings.
Step 2: Key Formula or Approach:
The work done in an adiabatic process from state 1 to state 2 is given by:
where this formula gives work done *by* the gas. For compression, the work is done *on* the gas, so .
We also use the adiabatic relation: .
Step 3: Detailed Explanation:
1. Determine Initial Conditions (State 1):
The gas is initially at NTP (Normal Temperature and Pressure). Mass, g. Density, g/cm³. Initial Volume, .
For NTP, we take dyn/cm². To match the answer options, it's common in such problems to approximate dyn/cm² (1 bar). Let's use this approximation.
dyn/cm².
2. Determine Final Conditions (State 2):
The gas is compressed to one-fourth of its original volume.
.
Using the adiabatic relation to find :
3. Calculate the Work Done:
We are calculating the work done *on* the gas (compression).
Substitute and :
Now, plug in the values:
Step 4: Final Answer:
The work done in compressing the air is ergs.

Step 1: Understanding the Concept:
The rate at which an object radiates thermal energy is described by the Stefan-Boltzmann law. The total energy radiated over a period of time is the power (rate of energy) multiplied by the time.
Step 2: Key Formula or Approach:
The Stefan-Boltzmann law for the power (P) radiated by a real object (not a perfect black body) is:
where is the emissivity (relative emittance), is the Stefan-Boltzmann constant, A is the surface area, and T is the absolute temperature.
The total energy (E) radiated in a time interval (t) is:
Step 3: Detailed Explanation:
1. Identify the given values:
Emissivity, .
Stefan's constant, .
Surface area, .
Temperature, .
Time, .
2. Calculate the radiated power (P):
3. Calculate the total energy (E) in one minute:
There seems to be a discrepancy between the calculated answer and the options provided. Let's re-examine the options. Note that . This suggests that the intended power calculation should have resulted in 27.36 W. This would happen if a slightly different value for Stefan's constant was used (approx ). Assuming there is a typo in the provided constant and the intended power is 27.36 W:
Assumed Power, .
Energy in 60 seconds, . This matches option (D) exactly. Given the structure of the options, this is the most likely intended answer.
Step 4: Final Answer:
Assuming the intended power output is 27.36 W due to a likely typo in the constants provided, the total energy radiated in one minute is 1641.6 J.

Concept: A reversible process is an ideal thermodynamic process that occurs in such a way that the system remains in equilibrium at every stage.
Step 1: Understanding reversible process.
For a process to be reversible, it must proceed infinitely slowly so that the system remains in thermodynamic equilibrium.
Step 2: Definition of quasi-static process.
A quasi-static process is one that occurs very slowly, passing through a sequence of equilibrium states.
Step 3: Connection to reversibility.
All reversible processes are quasi-static, as they require infinitesimally slow changes.
Step 4: Evaluating the options.

• Isothermal process Constant temperature, not necessarily slow (incorrect)
• Adiabatic process No heat exchange (incorrect)
• Quasi-static process Infinitesimally slow, reversible (correct)
• Isochoric process Constant volume (incorrect)

Step 5: Conclusion.
Thus, a quasi-static process must occur infinitesimally slowly to remain reversible.