Problem 15 - University Physics with Modern Physics

Question

A metal tank with volume 3.10 \(\mathrm{L}\) will burst if the absolute pressure of the gas it contains exceeds 100 atm. (a) If 11.0 mol of an ideal gas is put into the tank at a temperature of \(23.0^{\circ} \mathrm{C},\) to what temperature can the gas be warmed before the tank ruptures? You can ignore the thermal expansion of the tank. (b) Based on your answer to part (a), is it reasonable to ignore the thermal expansion of the tank? Explain.

Step-by-Step Solution

Answer

(a) 67.9°C; (b) Yes, ignoring thermal expansion is reasonable.

Step 1: Identify the Known Variables

We know the initial volume of the tank is \( V = 3.10 \, \text{L} = 3.10 \times 10^{-3} \, \text{m}^3 \), the number of moles \( n = 11.0 \), and the initial temperature \( T_1 = 23.0^{\circ} \text{C} = 296.15 \, \text{K} \). The maximum pressure \( P = 100 \, \text{atm} = 10132500 \, \text{Pa} \). Also, the gas constant \( R = 8.314 \, \text{J/mol} \cdot \text{K} \).

Step 2: Use the Ideal Gas Law to Find the Final Temperature

The ideal gas law is given by \( PV = nRT \). We need to find the temperature \( T_2 \) at which the gas pressure reaches 100 atm. Solving for \( T \), we have:\[ T_2 = \frac{PV}{nR} \]Substitute the values:\[ T_2 = \frac{(10132500 \, \text{Pa})(3.10 \times 10^{-3} \, \text{m}^3)}{(11.0 \, \text{mol})(8.314 \, \text{J/mol} \cdot \text{K})} \approx 341.05 \, \text{K} \]

Step 3: Convert the Temperature back to Celsius

To convert the temperature from Kelvin to Celsius, use the formula:\[ T_2(^{\circ}\text{C}) = T_2(\text{K}) - 273.15 \]Substitute the value of \( T_2 \):\[ T_2(^{\circ}\text{C}) = 341.05 - 273.15 = 67.9^{\circ}\text{C} \]

Step 4: Analyze the Thermal Expansion Consideration

A tank will expand with temperature, slightly increasing its volume and reducing internal pressure for the same gas amount. However, up to \( 67.9^{\circ}\text{C} \), if the tank material is strong and expansible, the theoretical calculations ignoring thermal expansion are generally close enough for ideal conditions. Hence, ignoring thermal expansion of the tank seems reasonable.

Key Concepts

Thermal Expansion

Thermal expansion describes how materials change in size or volume when they experience changes in temperature. In the context of gases, this typically involves heated molecules moving around more and occupying more space.
In solids and liquids, particles expand but less significantly than gases. When a metal tank heats up, it can expand slightly, increasing the space available inside.
For practical purposes, this expansion is often ignored when calculating the behavior of gases inside a tank, unless very high precision is needed. At higher temperatures, ignoring expansion may lead to small inaccuracies, but usually, it is considered negligible for rough calculations.

Absolute Pressure

Absolute pressure is the total pressure measured on a system, including atmospheric pressure. It differs from gauge pressure, which only measures the pressure exerted above atmospheric pressure. For example, in this exercise, the given absolute pressure limit is 100 atm, which translates to 10132500 Pa given that 1 atm equals 101325 Pa.
Understanding absolute pressure is crucial because it influences calculations involving gases. It includes all forms of pressure acting on a gas within a container. Ensuring the pressures are properly converted and understood allows for adequate analysis and predictions of potential ruptures and behavior of gases inside containers.

Temperature Conversion

Temperature conversions are necessary when different temperature scales are involved. For scientific calculations, it often involves converting degrees Celsius to Kelvin. Here's why:

• The Kelvin scale starts at absolute zero, making it suitable for physical calculations.
• Conversion between Celsius and Kelvin is straightforward:
  \[ T( ext{K}) = T(^ ext{C}) + 273.15 \]
• In this exercise, the temperature was originally at 23.0°C and needed to be in Kelvin for the equation. Converting gave 296.15 K.

Avoiding conversion mistakes helps ensure proper use of formulas and accurate results.

Volume Calculation

Understanding how to calculate the volume of gas is vital in thermodynamics. The initial volume was given as 3.10 liters, which equals 3.10 x 10^-3 cubic meters when converted. Using the ideal gas law, Volume (V) is essential in determining how changes in pressure, temperature, or amount of gas affect the system.
For calculations:

• Use the ideal gas law, \[ PV = nRT \], where V is volume, P is pressure, n is moles, R is gas constant, and T is temperature.
• This determines how the gas behaves under specific conditions.
• Volume conversions such as from liters to cubic meters ensure the formula units are consistent, which is pivotal for accuracy.

Through this understanding, volume becomes a facilitator for predicting gas properties in a controlled environment.