Problem 20 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

With what minimum pressure must a given volume of an ideal gas \((\gamma=1.4)\), originally at \(400 \mathrm{~K}\) and \(100 \mathrm{kPa}\), be adiabatically compressed in order to raise its temperature up to \(700 \mathrm{~K}\) ? (a) \(708.9 \mathrm{kPa}\) (b) \(362.5 \mathrm{kPa}\) (c) \(1450 \mathrm{kPa}\) (d) \(437.4 \mathrm{kPa}\)

Step-by-Step Solution

Answer

The minimum pressure needed is approximately \(708.9 \mathrm{kPa}\).

Step 1: Identify the Process

The process described is an adiabatic compression of an ideal gas. An adiabatic process is one where no heat is transferred to or from the gas. The relationship between pressure, volume, and temperature for an adiabatic process in an ideal gas is given by the equation \( P_1 V_1^{\gamma} = P_2 V_2^{\gamma} \) and \( \frac{T_1}{T_2} = \left(\frac{V_2}{V_1}\right)^{\gamma - 1} \), where \(\gamma\) is the heat capacity ratio.

Step 2: Calculate the Final Pressure

We can combine these equations to eliminate volume, using \( \frac{P_1}{P_2} = \left(\frac{T_1}{T_2}\right)^{\frac{\gamma}{\gamma-1}} \). We are given the initial pressure \(P_1 = 100 \mathrm{kPa}\), the initial temperature \(T_1 = 400 \mathrm{K}\), the final temperature \(T_2 = 700 \mathrm{K}\), and \(\gamma = 1.4\). Substituting these values yields \( P_2 = P_1 \left(\frac{T_2}{T_1}\right)^{\frac{\gamma}{\gamma - 1}} \).

Step 3: Compute the Minimum Pressure

Using the given values to find \(P_2\), we get \( P_2 = 100 \mathrm{kPa} \left(\frac{700 \mathrm{K}}{400 \mathrm{K}}\right)^{\frac{1.4}{1.4 - 1}} \). Solving this equation gives us the minimum pressure required to raise the temperature to \(700 \mathrm{~K}\) upon adiabatic compression.

Step 4: Solve the Equation

Now, calculate the actual value: \( P_2 = 100 \times \left(\frac{700}{400}\right)^{\frac{1.4}{0.4}} = 100 \times \left(\frac{7}{4}\right)^{3.5} \). By calculating the power and multiplying by 100, we will get the final answer for \(P_2\).

Step 5: Final Step

The calculation yields \( P_2 \approx 708.9 \mathrm{kPa} \), which matches with one of the answer choices provided.

Key Concepts

Ideal Gas Law

The ideal gas law is a fundamental equation in physics and chemistry that relates the pressure, volume, and temperature of an ideal gas. In its simplest form, the law is expressed as \( PV = nRT \), where \(P\) represents the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the universal gas constant, and \(T\) stands for temperature in Kelvin.

This equation is pivotal in solving problems related to gases under various conditions, and it serves as the cornerstone for understanding how gases behave in different situations. For an ideal gas, the particles are considered to be point particles with no volume and no interactions with each other, which, of course, is an approximation but serves well for many practical purposes.

In the context of adiabatic compression, the ideal gas law helps frame the relationship between the changing parameters of state when the gas is compressed without the exchange of heat with its surroundings.

Heat Capacity Ratio

The heat capacity ratio, often denoted by \(\gamma\) (gamma) and sometimes called the adiabatic index, is the ratio of the heat capacity at constant pressure \(C_p\) to the heat capacity at constant volume \(C_v\), or \(\gamma = \frac{C_p}{C_v}\).

This dimensionless quantity plays a crucial role in thermodynamics, particularly in adiabatic processes for an ideal gas. For monoatomic gases, \(\gamma\) is typically about 1.67, and for diatomic gases, like nitrogen or oxygen at room temperature, it's around 1.4, which is the value given in the problem.

The heat capacity ratio affects how the pressure and temperature of a gas change during an adiabatic process. In the context of the given exercise, \(\gamma\) is used to determine how much the pressure must increase to achieve a certain temperature rise during adiabatic compression, which is distinct from an isothermal process where the temperature would remain constant.

Thermodynamics

Thermodynamics is the branch of physics that deals with the relationships between heat and other forms of energy. In particular, it studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale. The laws of thermodynamics are universal, dictating the principles behind energy conversion and the directionality of processes.

An adiabatic process, a concept touched upon in the original exercise, is one of many types of thermodynamic processes and is defined as one in which no heat is exchanged with the surroundings. In terms of adiabatic compression, it's a process where work is done on the gas, increasing its pressure and temperature while maintaining the integrity of the system as insulated, preventing any heat transfer.

Understanding thermodynamics and its principles is essential for solving problems related to energy transfer and for grasping the underlying mechanisms of how energy transformations govern the physical world, including everything from engines to weather systems.