Question

In some of the thermodynamic cycles discussed in this chapter, one isotherm intersects one adiabatic curve. For an ideal gas, by what factor is the adiabatic curve steeper than the isotherm?

Short answer

Answer: The adiabatic curve is V^(γ-1) / γ times steeper than the isotherm for an ideal gas, where V is volume and γ is the adiabatic index (ratio of specific heat capacities, γ = C_P/C_V).

Step-by-step solution

Write down the equations for an adiabatic and isothermal process

For an ideal gas, the adiabatic process follows the equation: PV^γ = constant where P is pressure, V is volume, and γ is the adiabatic index (ratio of specific heat capacities, γ = C_P/C_V). For an isothermal process, the equation is given by: PV = constant

Differentiate both equations with respect to volume

Now, we can differentiate both equations with respect to volume (V) to find the slopes of the curves. For the adiabatic process: d(PV^γ)/dV = 0 Using the product rule, we get: Pd(V^γ)/dV + V^γ dP/dV = 0 For the isothermal process: d(PV)/dV = 0 Applying the product rule, we get: PdV/dV + VdP/dV = 0

Solve for dP/dV in both cases

For the adiabatic process: dP/dV = -[P(V^γ)]/(γV^(γ-1)) For the isothermal process: dP/dV = -P/V

Determine the factor by which the adiabatic curve is steeper

To find the factor by which the adiabatic curve is steeper, we can compare the magnitudes of the slopes. Divide the slope of the adiabatic curve by the slope of the isothermal curve. Factor = (dP/dV_adiabatic) / (dP/dV_isothermal) Factor = (-[P(V^γ)]/(γV^(γ-1))) / (-P/V) Factor = V^(γ-1) / γ The adiabatic curve is V^(γ-1) / γ times steeper than the isotherm for an ideal gas.

Key concepts

Adiabatic Process

When studying thermodynamics, it is crucial to understand different types of processes that can occur within a system. An adiabatic process is characterized by the absence of heat transfer into or out of the system. This means as the system does work, or work is done on it, its internal energy changes without exchanging heat with its environment.

For an ideal gas, this process is governed by the equation \(PV^{\gamma} = \text{constant}\), where \(P\) stands for pressure, \(V\) for volume, and \(\gamma\) is known as the adiabatic index. The adiabatic index is the ratio of the specific heats at constant pressure and constant volume, \(C_P/C_V\). A key attribute of an adiabatic process is that it can be reversible or irreversible, depending on whether the system is isolated from or interacts with its surroundings in ways other than heat transfer.

Differentiating the adiabatic equation with respect to volume allows us to find the rate at which pressure changes with volume. This rate, or slope, indicates how steep the change in pressure with volume will be on a graph representing the system's state - a steeper slope signifies a more rapid change in pressure for a given change in volume.

Isothermal Process

An isothermal process occurs when a system exchanges heat with its surroundings to maintain a constant temperature. For an ideal gas, this process can be described with the simple equation \(PV = \text{constant}\) - also known as Boyle's Law. This relationship reflects how the pressure and volume of a gas are inversely proportional when temperature is held steady.

During an isothermal expansion or compression, heat either enters or leaves the system to ensure that its temperature doesn't change. In practical terms, it implies that the system is perfectly in thermal equilibrium with its surroundings. To visualize this, if we graph pressure against volume, the line we observe is called an isotherm. In the context of isothermal processes for ideal gases, the concept of the ideal gas laws comes into play, where relationships among pressure, volume, and temperature are simplified thanks to the ideal gas assumption which negates interactions between molecules.

Rather than producing work through the change in internal energy as in adiabatic processes, the work done by or on an isothermal system results purely from energy transfer as heat.

Ideal Gas

The concept of an ideal gas is a simplified model that allows us to understand gas behaviors under various conditions without delving into the complexity of real-world interactions. An ideal gas is a hypothetical gas that perfectly follows the ideal gas law, \(PV=nRT\), where \(P\) is the pressure, \(V\) is the volume, \(n\) is the amount of gas in moles, \(R\) is the universal gas constant, and \(T\) is the temperature in Kelvin.

The ideal gas approximation assumes that the gas particles are infinitesimally small, do not attract or repel each other, and are in constant, random motion. Collisions between these particles and with the walls of their container are perfectly elastic, meaning there is no loss of kinetic energy. In reality, no gas is genuinely ideal, but the ideal gas law provides an excellent approximation for many gases under a range of conditions.

In the context of comparing isothermal and adiabatic processes for an ideal gas, as in the original textbook exercise, this simplification is invaluable. It allows us to focus on the process while minimizing distractions from complex intermolecular forces or changes in internal energy beyond the scope of the process we're examining.