Thermodynamics

Power and Heat Removed Calculation

The equation for the coefficient of performance ( ) is given by:

Where: - is the heat removed, - is the work done, - and are the temperatures at the hot and cold reservoirs, respectively.

Step-by-Step Calculation:

Substitute the given values into the equation:

Simplifying the right-hand side:

Now, solve for :

Step 2: Calculate Power

Next, we calculate the power using the formula:

Conclusion:

The calculated power is .

Concept:
When a gas is compressed, the work done depends on the nature of the thermodynamic process.

Isothermal Process:
In an isothermal process, the temperature remains constant. The work done in compressing a gas from volume to is:


Adiabatic Process:
In an adiabatic process, there is no heat exchange. The work done is:

or derived using:

Key Point: In adiabatic compression, temperature rises, so internal energy increases → more work is required.

Since both compressions reduce the volume to half, we compare them:

• Isothermal: Temperature constant → some heat leaves → less work needed.
• Adiabatic: No heat exchange → temperature increases → internal energy increases → more work required.

Correct Answer:

Isochoric: It is a thermodynamic process that occurs at a constant volume. During this process the work done on or by the system is zero. Therefore it is path I
Adiabatic: It is a thermodynamic process in which there is no exchange of heat between the system and its surrounding. Therefore it is path II Isothermal: It is a thermodynamic process that occurs at a constant temperature. Therefore it is path III Isobaric: It is a thermodynamic process that occurs at a constant pressure. Therefore it is path IV.

In the context of thermodynamics, an ideal gas can undergo several types of processes, each defined by how certain parameters change:

• Adiabatic Process: No heat exchange occurs (Q = 0). The process is characterized by a steeper curve on a PV diagram compared to an isothermal process.
• Isothermal Process: Temperature remains constant (ΔT = 0). The PV curve follows a hyperbola, defined by PV = nRT.
• Isobaric Process: Pressure remains constant (ΔP = 0). The PV curve is a horizontal line.
• Isochoric Process: Volume remains constant (ΔV = 0). The PV curve is a vertical line.

In the provided figure (unavailable for reference here), we need to identify the curve representing the adiabatic process. Generally, an adiabatic process in an ideal gas follows the equation:

P*V^γ = constant

where γ (gamma) is the heat capacity ratio (C_p/C_v). The curve for the adiabatic process is steeper compared to the isothermal curve (because γ > 1 for ideal gases).

Given the provided options:

• A: Typically less steep
• B: The steepest curve
• C: A likely candidate for isothermal (flatter curve)
• D: A likely candidate for isobaric or isochoric

The process that appears to be the steepest, and thus corresponds to the adiabatic process, is option B.

Therefore, the adiabatic process is represented by curve 2.

The efficiency of a Carnot engine is given by the formula:

η=1−(T_c/T_h)

where η is the efficiency, T_c is the temperature of the cold reservoir (sink), and T_h is the temperature of the hot reservoir (source). All temperatures are in Kelvin. Given that the efficiency η is 50%, this can be represented as 0.5. The temperature of the hot reservoir T_h is 327°C, which converts to Kelvin as follows:

T_h=327+273=600 K

Substituting the values in the efficiency equation:

0.5=1−(T_c/600)

Simplifying the equation gives:

0.5=(T_c/600)

Solve for T_c:

T_c=0.5×600=300 K

Convert the temperature back to Celsius:

T_c=300−273=27°C

Thus, the temperature of the sink is 27°C.

Understanding the Concept

Work done by a gas in a thermodynamic process is given by the integral of pressure with respect to volume:

In simpler terms, on a P-V diagram, the work done is represented by the area under the curve for the given process.

Analyzing Path bc

Looking at the P-V diagram, path bc is a vertical line. This means the volume remains constant during this process. Such a process is called an isochoric or isovolumetric process.

Since the volume doesn't change (dV = 0), the work done along this path is:

Therefore, the work done by the gas along the path bc is 0.

For a polyatomic gas with 3 translational, 3 rotational, and vibrational modes:

Internal energy (U)