Problem 70 - University Physics with Modern Physics

Question

A gas expands at constant pressure from \(3.00 \mathrm{~L}\) at \(15.0^{\circ} \mathrm{C}\) until the volume is \(4.00 \mathrm{~L}\). What is the final temperature of the gas?

Step-by-Step Solution

Answer

Answer: The final temperature of the gas is \(384.2 \mathrm{~K}\).

Step 1: Convert temperatures to Kelvin

First, we need to convert the given initial temperature from Celsius to Kelvin using the formula: K = °C + 273.15 Initial temperature = \(15 ^\circ \mathrm{C}\) \(T_1 = 15 + 273.15 = 288.15 \mathrm{~K}\)

Step 2: Set up Charles' Law formula

Using the Charles' Law formula for gas under constant pressure: \(\frac{V_1}{T_1} = \frac{V_2}{T_2}\) Given: \(V_1 = 3.00 \mathrm{~L}\), \(V_2 = 4.00 \mathrm{~L}\), and \(T_1 = 288.15 \mathrm{~K}\) Our goal is to find the final temperature, \(T_2\).

Step 3: Substitute the values and solve for the final temperature

Now we'll substitute the given values into the formula and solve for the final temperature: \(\frac{3.00 \mathrm{~L}}{288.15 \mathrm{~K}} = \frac{4.00 \mathrm{~L}}{T_2}\) To solve for \(T_2\), we'll cross-multiply and divide: \(T_2 = \frac{4.00 \mathrm{~L} \cdot 288.15 \mathrm{~K}}{3.00 \mathrm{~L}}\)

Step 4: Calculate the final temperature

Now, we'll perform the calculations: \(T_2 = \frac{4.00 \cdot 288.15}{3.00}\) \(T_2 = 384.2 \mathrm{~K}\) The final temperature of the gas is \(384.2 \mathrm{~K}\).