Problem 16 - University Physics with Modern Physics

Question

Three moles of an ideal gas are in a rigid cubical box with sides of length 0.200 \(\mathrm{m}\) . (a) What is the force that the gas exerts on each of the six sides of the box when the gas temperature is \(20.0^{\circ} \mathrm{C} ?\) (b) What is the force when the temperature of the gas is increased to \(100.0^{\circ} \mathrm{C}\) ?

Step-by-Step Solution

Answer

The force at 20°C is approx. 36,609.365 N; at 100°C, it is approx. 46,636.133 N.

Step 1: Understand the Problem

We need to find the force the gas exerts on each side of the box at given temperatures and using the ideal gas law.

Step 2: Apply the Ideal Gas Law

The ideal gas law is \[ PV = nRT \]where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant \((8.314 \, J/(mol \cdot K))\), and \( T \) is the temperature in Kelvin. The volume \( V \) of the box is given by \( V = 0.2^3 = 0.008 \, m^3 \).

Step 3: Convert Temperatures to Kelvin

Convert the given temperatures from Celsius to Kelvin using the formula \( T(K) = T(°C) + 273.15 \). - At \( 20.0^{\circ} \mathrm{C} \), \( T = 293.15 \, K \).- At \( 100.0^{\circ} \mathrm{C} \), \( T = 373.15 \, K \).

Step 4: Calculate Pressure at 20.0°C

Substitute the values into the ideal gas law to solve for pressure \( P \) at \( T = 293.15 \, °K \):\[ P = \frac{nRT}{V} = \frac{(3 \, \text{moles})(8.314 \, J/(mol \cdot K))(293.15 \, K)}{0.008 \, m^3} \]\[ P \approx 915,234.125 \, N/m^2 \].

Step 5: Calculate Force on Each Side at 20.0°C

Use the formula \( F = PA \) where \( A \) is the area of one side of the box, which is \( 0.2^2 = 0.04 \, m^2 \). Thus, the force is:\[ F = 915,234.125 \, N/m^2 \times 0.04 \, m^2 \]\[ F \approx 36,609.365 \, N \].

Step 6: Calculate Pressure at 100.0°C

Use the same steps to find pressure at \( T = 373.15 \, K \):\[ P = \frac{(3)(8.314)(373.15)}{0.008} \] \[ P \approx 1,165,903.3125 \, N/m^2 \].

Step 7: Calculate Force on Each Side at 100.0°C

Calculate the force at the new pressure:\[ F = 1,165,903.3125 \, N/m^2 \times 0.04 \, m^2 \]\[ F \approx 46,636.1325 \, N \].

Key Concepts

Pressure Calculation

When calculating the pressure exerted by a gas, we rely on the ideal gas law: \[ PV = nRT \] Here, \( P \) denotes pressure, \( V \) is the volume, \( n \) represents the number of moles, \( R \) is the ideal gas constant, and \( T \) refers to temperature in Kelvin.
In our example, we want to find the pressure the gas exerts within a rigid cubical box with known dimensions. The key steps to calculate this pressure include:

• Determining the volume of the cube using the formula \( V = \, {\text{side length}}^3 \).
• Substituting the known values into the ideal gas law to solve for pressure \( P \).

As temperature or the number of moles changes, pressure will adjust according to the ideal gas equation.

Temperature Conversion

Temperature conversion is crucial because the ideal gas law requires temperatures to be in Kelvin rather than Celsius.
The conversion process involves a simple formula: \[ T(K) = T(°C) + 273.15 \] This formula adds 273.15 to the Celsius temperature to convert it to Kelvin.
In the given problem, the temperatures provided are:\( 20.0^{\circ} \mathrm{C} \) and \( 100.0^{\circ} \mathrm{C} \).

• Converting \( 20.0^{\circ} \mathrm{C} \) to Kelvin: \( T = 20 + 273.15 = 293.15 \, K \).
• Converting \( 100.0^{\circ} \mathrm{C} \) to Kelvin: \( T = 100 + 273.15 = 373.15 \, K \).

This conversion ensures that our calculations using the ideal gas law are consistent and accurate.

Force Exerted by Gas

Once we find the pressure, next is to determine the force exerted by the gas on the box walls. This force can be calculated by using the relation:\[ F = PA \] where \( F \) is the force, \( P \) is the pressure, and \( A \) is the area of the surface on which the force is acting.
In this scenario, \( A \) is the area of each side of the cube:
\( A = 0.2^2 = 0.04 \, m^2 \).

• For a pressure of \( 915,234.125 \, N/m^2 \) at \( 20.0^{\circ} \mathrm{C} \), the force is \( F \approx 36,609.365 \, N \).
• For a pressure of \( 1,165,903.3125 \, N/m^2 \) at \( 100.0^{\circ} \mathrm{C} \), the force is \( F \approx 46,636.1325 \, N \).

These calculations show how increased temperature results in higher pressure and, subsequently, more force exerted by the gas.

Volume of a Cube

Calculating the volume of a cube is a straightforward task, but very important for applying the ideal gas law. The volume \( V \) of a cube is found by raising the side length to the third power:\[ V = l^3 \] where \( l \) is the length of a side of the cube.
For a cube with a side length of \( 0.200 \, m \), the volume would be:\[ V = 0.2^3 = 0.008 \, m^3 \]
Understanding how to determine the cube's volume is essential, as it directly influences the application of the ideal gas law. The calculated volume is used as the \( V \) in the ideal gas equation, ensuring the accuracy of the pressure and force results.