Problem 32 - College Physics

Question

If \(2.4 \mathrm{~m}^{3}\) of a gas initially at STP is compressed to \(1.6 \mathrm{~m}^{3}\) and its temperature is raised to \(30^{\circ} \mathrm{C}\), what is its final pressure?

Step-by-Step Solution

Answer

The final pressure is approximately 1.67 atm.

Step 1: Understand the Problem

We are given an initial volume of gas at standard temperature and pressure (STP), which means a temperature of 0°C (273 K) and a pressure of 1 atm. We need to find the final pressure when the gas is compressed and warmed up.

Step 2: Convert Temperature

Convert the temperature from degrees Celsius to Kelvin because the ideal gas law requires temperature in Kelvin. The conversion formula is: \[ T(K) = T(°C) + 273\] For our problem: \[T = 30°C + 273 = 303 \text{ K}\]

Step 3: Apply the Combined Gas Law

The combined gas law is given by \[\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \]Where:- \(P_1\) = 1 atm (initial pressure)- \(V_1\) = 2.4 m³ (initial volume)- \(T_1\) = 273 K (initial temperature)- \(V_2\) = 1.6 m³ (final volume)- \(T_2\) = 303 K (final temperature)We need to solve for \(P_2\), the final pressure.

Step 4: Resolve the Equation for Final Pressure

Rearrange the combined gas law to solve for \(P_2\):\[P_2 = \frac{P_1 V_1 T_2}{V_2 T_1}\]Substitute the known values:\[P_2 = \frac{1 \times 2.4 \times 303}{1.6 \times 273}\]

Step 5: Calculate the Final Pressure

Perform the calculation:\[P_2 = \frac{2.4 \times 303}{1.6 \times 273} = \frac{727.2}{436.8} \approx 1.666\]So, the final pressure \(P_2\) is approximately 1.67 atm.

Key Concepts

Ideal Gas Law

The ideal gas law is a fundamental principle in chemistry that relates the pressure, volume, and temperature of an ideal gas using the formula: \[ PV = nRT \]- **P** represents pressure, measured in atmospheres (atm) or pascals (Pa). - **V** denotes the volume of the gas, often in liters (L) or cubic meters (m³).- **n** is the number of moles of gas.- **R** is the ideal gas constant, typically 0.0821 L·atm/mol·K.- **T** stands for temperature, which should always be in kelvin (K).
The ideal gas law provides a simple equation for calculating any one property of a gas if the others are known, assuming that the gas behaves ideally. While more generalized than the combined gas law, it is often used together in problem-solving to find unknown variables in gas law equations.

Temperature Conversion

When working with gas laws, especially the ideal and combined gas laws, it is crucial to convert temperatures into kelvin. Kelvin is the SI unit for temperature and is required for calculations involving gases. The conversion is simple: \[ T(K) = T(°C) + 273 \]This formula allows you to easily switch from Celsius, a more commonly used scale in everyday life, to Kelvin.
For example, if the temperature is 30°C, then:

• Convert 30°C to Kelvin: \[ 30 + 273 = 303 \] K

Remember, Kelvin is absolute, meaning it does not go below zero. So, this conversion ensures that all calculations with gas laws are accurate and meaningful.

Pressure Calculation

Calculating pressure changes in gases often involves using the combined gas law.This equation blends Boyle's Law, Charles's Law, and Gay-Lussac's Law effectively.The equation is:\[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \]It relates the initial and final states of a gas.Here, you can calculate the final pressure if you know the initial pressure, volumes, and temperatures.
In the given example:

• Initial pressure \( P_1 \) is 1 atm (at standard temperature and pressure).
• Initial volume \( V_1 \) is 2.4 m³.
• The initial temperature \( T_1 \) is 273 K.
• Final volume \( V_2 \) is 1.6 m³ and temperature \( T_2 \) is 303 K.
Rearranging the equation to solve for \( P_2 \), the final pressure:\[ P_2 = \frac{P_1 V_1 T_2}{V_2 T_1} \]Put the known values into this equation to get your answer.

Volume Compression

Volume compression refers to the reduction in the space a gas occupies. This physical change of state directly influences the gas pressure and temperature. According to Boyle's Law, volume and pressure are inversely proportional, meaning if you decrease the volume of a gas, its pressure increases, provided the temperature remains constant.
In the problem scenario given, volume compression plays a central role. The initial volume of the gas is 2.4 m³ and it is compressed to 1.6 m³.

• As a result of this compression, the volume reduces—and since temperature also changes—this affects the pressure.
• Understanding how volume compression affects gas properties helps you effectively apply the combined gas law.
• Always keep in mind the relationship: small volume = high pressure if the amount of gas and temperature are constant.

Such insights into volume compression guide how to approach problems using the combined gas law and ideal gas law effectively.