Question

A sample of hydrogen at 47°C exerts a pressure of 0.329 atm. The gas is heated to 77°C at constant volume. What will its new pressure be?

Short answer

The new pressure is approximately 0.359 atm.

Step-by-step solution

- Convert Temperatures to Kelvin

First, convert the given temperatures from Celsius to Kelvin. Use the formula: \[ T(K) = T(°C) + 273.15 \]For the initial temperature:\[ T_1 = 47 + 273.15 = 320.15 \text{ K} \]For the final temperature:\[ T_2 = 77 + 273.15 = 350.15 \text{ K} \]

- Apply Gay-Lussac's Law

Since the volume is constant, use Gay-Lussac's Law, which states that the pressure of a gas is directly proportional to its temperature (in Kelvin) when volume is constant:\[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \]Substitute the known values:\[ \frac{0.329 \text{ atm}}{320.15 \text{ K}} = \frac{P_2}{350.15 \text{ K}} \]

- Solve for the New Pressure

Rearrange the formula to solve for the new pressure \( P_2 \):\[ P_2 = P_1 \times \frac{T_2}{T_1} \]Substitute the known values:\[ P_2 = 0.329 \text{ atm} \times \frac{350.15 \text{ K}}{320.15 \text{ K}} \]Calculate the result:\[ P_2 \approx 0.359 \text{ atm} \]

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation that describes the behavior of gases. It combines several gas laws into one equation: \[ PV = nRT \]Here,

• P = Pressure of the gas
• V = Volume of the gas
• n = Amount of gas (in moles)
• R = Ideal gas constant (8.314 J/(mol·K))
• T = Temperature (in Kelvin)

The equation shows how pressure, volume, and temperature are related in an ideal gas. When dealing with such questions, remember:

• If one variable changes, others will adjust to satisfy this equation.
• It is crucial to always use Kelvin for temperature when applying this law.

This law is a cornerstone for understanding Gay-Lussac's Law, which specifically analyses the Pressure-Temperature relationship when volume is held constant.

Pressure-Temperature Relationship

Gay-Lussac's Law focuses on how pressure changes with temperature in a gas kept at constant volume. Mathematically, it is written as:\[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \]Where,

• P₁ is the initial pressure
• T₁ is the initial temperature (in Kelvin)
• P₂ is the final pressure
• T₂ is the final temperature (in Kelvin)

Gay-Lussac's Law tells us that if you increase the temperature of a gas, its pressure will increase proportionally—as long as the volume does not change. In the exercise, the temperature of the hydrogen gas changes from 47°C to 77°C:

• Initial temperature (in Kelvin) = 320.15 K
• Final temperature (in Kelvin) = 350.15 K

Since the gas is heated, the pressure increases, calculated using the earlier stated relationship. It is important to use Kelvin to get accurate results, as the Kelvin scale starts at absolute zero, ensuring all temperatures are positive.

Converting Celsius to Kelvin

To solve gas law problems accurately, you need to convert temperatures from Celsius to Kelvin. The Kelvin scale is the SI unit for temperature and provides consistent results in gas law calculations. The conversion formula is simple:\[ T(K) = T(°C) + 273.15 \]Here's how you convert Celsius to Kelvin:

• Example: Convert 47°C to Kelvin:
• Initial temperature (T₁): \[ 47 + 273.15 = 320.15 \text { K} \]
• Example: Convert 77°C to Kelvin:
• Final temperature (T₂):\[ 77 + 273.15 = 350.15 \text { K} \]

Make sure to always add 273.15 to the given Celsius temperature. This ensures you are using the correct temperature values to apply the Ideal Gas Law and Gay-Lussac's Law effectively.