Question

What will happen to the pressurc of a 5 litres sample of a gas at 5 atmospheres if it is heated from \(250 \mathrm{~K}\) to \(300 \mathrm{~K}\) and the volume is kept constant? (1) \(6 \mathrm{~atm}\) (2) \(4.16 \mathrm{~atm}\) (3) \(3 \mathrm{~atm}\) (4) \(2.0 \mathrm{~atm}\)

Short answer

(1) 6 atm

Step-by-step solution

Identify the appropriate gas law

Since the volume is kept constant, use Gay-Lussac's Law, which states that the pressure of a gas is directly proportional to its temperature when the volume is held constant. The formula is \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \).

Assign known values

Identify and assign the known values: \( P_1 = 5 \, \text{atm} \), \( T_1 = 250 \, \text{K} \), and \( T_2 = 300 \, \text{K} \). We'll solve for \( P_2 \).

Rearrange the formula to solve for the unknown pressure

Use the equation and rearrange to solve for \( P_2 \). \( P_2 = P_1 \cdot \frac{T_2}{T_1} \)

Substitute the known values

Insert the given values into the equation: \( P_2 = 5 \cdot \frac{300}{250} \)

Calculate the result

Simplify and solve the equation: \( P_2 = 5 \cdot 1.2 = 6 \, \text{atm} \)

Determine the correct answer

Compare the result with the choices given. The correct answer is \(6 \, \text{atm} \) which corresponds to option (1).

Key concepts

gas laws

Gas laws are essential principles that describe the behavior of gases in different conditions. These laws help us understand how changing one variable affects others, like pressure, volume, and temperature. The main gas laws include Boyle's Law, Charles's Law, and Gay-Lussac's Law.

Boyle's Law tells us that the pressure of a gas is inversely proportional to its volume when temperature remains constant. Charles's Law indicates that the volume of a gas is directly proportional to its temperature at constant pressure.

In this exercise, we focus on Gay-Lussac's Law.
This law states that the pressure of a gas is directly proportional to its temperature when its volume is held constant. The formula used is:
\[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \]
Here, \( P_1 \) and \( T_1 \) are the initial pressure and temperature, while \( P_2 \) and \( T_2 \) are the final pressure and temperature. Understanding these laws helps us predict how a gas will behave in various situations.

pressure-temperature relationship

The pressure-temperature relationship of a gas is crucial in various scientific and practical applications. Gay-Lussac's Law specifically addresses this relationship under constant volume. It states that as the temperature of a gas increases, the pressure also increases proportionally, provided the volume does not change.

For our problem, we start with an initial pressure \( P_1 \) of 5 atm and an initial temperature \( T_1 \) of 250 K. When the gas is heated to a new temperature \( T_2 \) of 300 K, the volume remains constant. According to the formula:
\[ P_2 = P_1 \cdot \frac{T_2}{T_1} \]
We substitute the given values into the equation and calculate the final pressure \( P_2 \). This relationship helps validate and predict outcomes in various scenarios like the efficiency of car engines and understanding weather patterns.

If you increase the temperature of a confined gas, its particles move faster and collide more often with the walls of its container. These frequent collisions result in increased pressure.

ideal gas law applications

The ideal gas law combines all the individual gas laws into one comprehensive equation:
\[ PV = nRT \]
Here, \( P \) represents pressure, \( V \) is volume, \( n \) denotes the number of moles, \( R \) is the ideal gas constant, and \( T \) stands for temperature. While this exercise specifically focuses on Gay-Lussac's Law due to constant volume conditions, the ideal gas law can also help solve many other gas-related problems.

Applications of the ideal gas law are numerous and varied. For example:

• It helps in calculating the amount of gas needed for a specific reaction in chemistry.
• Predicts the behavior of gases under different temperatures and pressures, which is crucial in fields like engineering and meteorology.
• It provides insights into natural phenomena, like how temperature and pressure changes contribute to weather patterns.

This law is foundational in understanding the properties and behavior of gases in everyday and industrial contexts.