Question

Which of the statement is correct? (a) Slope of adiabatic \(\mathrm{P}-\mathrm{V}\) curve will be same as that of isothermal one. (b) Slope of adiabatic \(\mathrm{P}-\mathrm{V}\) curve is smaller than that in isothermal one (c) Slope of adiabatic \(\mathrm{P}-\mathrm{V}\) curve is larger than that in isothermal one. (d) Slope of adiabatic \(\mathrm{P}-\mathrm{V}\) curve may be smaller or larger depending on the value \(\mathrm{V}\).

Short answer

(c) The slope of the adiabatic \( P-V \) curve is larger than that in the isothermal one.

Step-by-step solution

Understanding the Concept

Both adiabatic and isothermal processes describe different thermodynamic transformations. An isothermal process occurs at a constant temperature, while an adiabatic process occurs without heat exchange, meaning the temperature changes as the system evolves.

Review of Mathematical Expressions

For an isothermal process, we have the equation: \( PV = ext{constant} \). The slope for an isothermal curve in a \( P-V \) diagram is \( \frac{dP}{dV} = -\frac{P}{V} \). For an adiabatic process, the equation is \( PV^\gamma = ext{constant} \), where \( \gamma \) is the heat capacity ratio (\( C_p/C_v \)). The slope for an adiabatic process is \( \frac{dP}{dV} = -\gamma\frac{P}{V} \).

Comparing Slopes

By comparing the slopes, we see that: \(-\gamma\frac{P}{V} > -\frac{P}{V} \) because \( \gamma \) (which is typically greater than 1 for gases) increases the magnitude of the slope. This shows that the slope of an adiabatic curve is steeper (or larger in magnitude) than that of an isothermal curve.

Drawing the Conclusion

Since the adiabatic slope \(-\gamma\frac{P}{V} \) is steeper than the isothermal slope \(-\frac{P}{V} \), we can conclude that the adiabatic curve's slope is larger.

Key concepts

Adiabatic Process

An adiabatic process is a thermodynamic transformation that occurs without any heat exchange between the system and its surroundings. This means that all the energy changes occur within the system itself, adjusting in form between work and the internal energy, without the system gaining or losing heat from the outside environment.
In an adiabatic process, any work done by or on the system results in a change in the internal energy, which often leads to a change in temperature. For example, a gas being compressed adiabatically will experience a rise in temperature because the work done on it increases its internal energy. Similarly, when a gas expands adiabatically, it cools down as its internal energy decreases to do the work of expansion. Key characteristics of an adiabatic process:

• No heat transfer: The system is perfectly insulated from its surroundings.
• Changes in internal energy lead to temperature changes.
• Typical in rapid processes where there is no time for heat transfer, such as in rapid compression or expansion, and in insulated systems.

These processes are governed by the equation: \[ PV^\gamma = \text{constant} \]where \( \gamma \) is the heat capacity ratio \( C_p/C_v \), signifying how the pressure and volume are interdependent.

Isothermal Process

An isothermal process, in contrast, is a process where the temperature remains constant throughout. That requires a perfect balance of heat flow into or out of the system to offset any changes in energy that would otherwise change temperature. When a gas is compressed isothermally, heat is removed to maintain constant temperature. Conversely, when gas expands isothermally, heat is added. For an isothermal process involving an ideal gas, the work done can be calculated using the equation:\[ W = nRT \ln \left(\frac{V_f}{V_i}\right) \]where \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the absolute temperature. Characteristics of an isothermal process:

• Temperature is constant: The system exchanges heat with its surrounding to maintain an unchanging temperature.
• Typically slower processes, allowing time for heat exchange.
• Described by the equation: \( PV = \text{constant} \), reflecting Boyle’s Law.

It's common in processes that occur in thermal equilibrium, where the system's temperature matches that of its surroundings.

P-V Diagram

The Pressure-Volume (P-V) diagram is a graphical representation of the changes in pressure and volume in a thermodynamic process. It is highly useful for illustrating how different processes relate to each other, such as adiabatic and isothermal processes. Each curve on a P-V diagram presents the relationship between pressure and volume for a system undergoing a particular process. On the P-V diagram, an adiabatic process and an isothermal process appear as distinct curves:

• The isothermal curve (or isotherm) is typically less steep, reflecting the constant temperature condition and the equation \( PV = \text{constant} \).
• The adiabatic curve is steeper, as described by the equation \( PV^\gamma = \text{constant} \), and it indicates a temperature change as no heat is exchanged.

In this context, understanding the slope differences matters:

• For isothermal processes, the slope is \(-\frac{P}{V}\).
• For adiabatic processes, the slope is \(-\gamma \frac{P}{V}\), steeper due to \( \gamma > 1 \).

These differences in slopes help visualize the dynamics of energy exchange—whether constant as heat flow in isothermal or changing as internal energy in adiabatic processes. Thus, the P-V diagram serves as a powerful tool to interpret and differentiate these thermodynamic processes.