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

(a) 36580.47 N; (b) 46595.87 N.

Step 1: Calculate Volume of the Box

The volume of a cube with side length 0.200 m is given by the formula \( V = L^3 \). Substituting the side length, we have: \[ V = (0.200 \, \text{m})^3 = 0.008 \, \text{m}^3 \].

Step 2: Convert Temperatures to Kelvin

Since we are dealing with ideal gases, we need to use Kelvin for temperature. Convert \(20.0^{\circ} \mathrm{C}\) to Kelvin using the formula \( T(K) = T(^{\circ}\mathrm{C}) + 273.15 \):\[ T_1 = 20 + 273.15 = 293.15 \, \text{K} \].\[ T_2 = 100 + 273.15 = 373.15 \, \text{K} \].

Step 3: Use Ideal Gas Law to Find Pressure at 20°C

The ideal gas law states \( PV = nRT \). For this step, solve for pressure at \(20^{\circ}C\):\[ P_1 = \frac{nRT_1}{V} \]. Substitute \( n = 3 \), \( R = 8.314 \, \text{J/mol·K} \), \( T_1 = 293.15 \, \text{K} \), and \( V = 0.008 \, \text{m}^3 \): \[ P_1 = \frac{3 \times 8.314 \times 293.15}{0.008} = 914511.73 \, \text{Pa} \].

Step 4: Calculate Force on Each Side at 20°C

The force exerted by the gas on each side of the box is given by \( F = P \times A \), where \( A \) is the area of one side of the cube: \( A = (0.200 \, \text{m})^2 = 0.04 \, \text{m}^2 \).\[ F_1 = 914511.73 \times 0.04 = 36580.47 \, \text{N} \].

Step 5: Use Ideal Gas Law to Find Pressure at 100°C

Calculate pressure at \(100^{\circ}C\):\[ P_2 = \frac{nRT_2}{V} \]. Substitute \( n = 3 \), \( R = 8.314 \, \text{J/mol·K} \), \( T_2 = 373.15 \, \text{K} \), and \( V = 0.008 \, \text{m}^3 \): \[ P_2 = \frac{3 \times 8.314 \times 373.15}{0.008} = 1164896.79 \, \text{Pa} \].

Step 6: Calculate Force on Each Side at 100°C

Use the same area as before to find the force when the gas temperature is \(100^{\circ}C\):\[ F_2 = 1164896.79 \times 0.04 = 46595.87 \, \text{N} \].

Key Concepts

Pressure Calculation

To calculate pressure in a system involving ideal gases, the ideal gas law is commonly used. This law is represented by the equation \( PV = nRT \). Here, \( P \) stands for pressure, \( V \) is volume, \( n \) represents the number of moles of the gas, \( R \) is the ideal gas constant (8.314 J/mol·K), and \( T \) is the temperature in Kelvin.
When solving for pressure, rearrange the formula as \( P = \frac{nRT}{V} \). This allows us to determine the pressure exerted by the gas based on the given conditions.
The values for these variables are typically given in the problem or can be calculated, such as volume from the dimensions of a container. The result yields the pressure in Pascals (Pa), a commonly used unit in scientific calculations of pressure.

Temperature Conversion

When working with the ideal gas law, it's important to convert temperatures from degrees Celsius (°C) to Kelvin (K). This is because Kelvin is the standard temperature unit for scientific calculations involving ideal gases.
The conversion from Celsius to Kelvin is straightforward: you simply add 273.15 to the temperature in Celsius.

• For example, to convert \(20.0^{\circ} \mathrm{C}\) to Kelvin, use the formula \( T(K) = 20 + 273.15 \), which equals 293.15 K.
• Similarly, for \(100.0^{\circ} \mathrm{C}\), the conversion is \( T(K) = 100 + 273.15 \), resulting in 373.15 K.

This conversion allows for accurate usage in equations of state like the ideal gas law.

Force Exertion

The force exerted by a gas on the walls of its container can be calculated using the formula \( F = P \times A \), where \( F \) is force, \( P \) is pressure, and \( A \) is the area of the wall.
This concept is essential in understanding how gases interact with their surroundings. For a cube, with each side having an area \( A \) that is calculated as side length squared, the formula becomes:

• If the side length is 0.200 m, the area \( A = (0.200 \, \text{m})^2 = 0.04 \, \text{m}^2 \).

By substituting the values for pressure calculated via the ideal gas law, you can determine the force a gas imposes on each side of the container at any given temperature.

Ideal Gas Concepts

Ideal gas concepts involve understanding how gases behave when they follow the theoretical model of the ideal gas law. This model assumes no interactions between molecules and that the molecules occupy no volume themselves. While real gases do experience these interactions and have volume, under many conditions, they can be approximated as ideal gases.
In problems involving ideal gases, key parameters include:

• **Pressure (P):** The force exerted by gas particles colliding with container walls.
• **Volume (V):** The space the gas occupies.
• **Number of moles (n):** Quantity of gas, which impacts pressure if conditions change.
• **Temperature (T):** Affects the kinetic energy and thus the pressure exerted by the gas.

Analyzing how these parameters interact helps in predicting gas behavior under different conditions. Understanding these concepts is crucial for accurate problem-solving and grasping further gas-related phenomena.