Question

Argon gas is enclosed in a container of a fixed volume. At \(T=300 \mathrm{~K}\), the pressure of the gas is \(50,000 \mathrm{~Pa}\). At \(T=350 \mathrm{~K},\) calculate the pressure of the gas.

Short answer

The pressure of the gas at \(T=350\,\mathrm{K}\) is approximately \(58,333.33\,\mathrm{Pa}\).

Step-by-step solution

Identify Given Variables

Identify the given variables from the problem statement. The initial temperature, \(T_1 = 300\,\mathrm{K}\), the initial pressure, \(P_1 = 50,000\, \mathrm{Pa}\), and the final temperature, \(T_2 = 350\, \mathrm{K}\) are provided. We need to find the final pressure, \(P_2\).

Apply the Ideal Gas Law for Fixed Volume

Since the volume is constant, the relationship between pressure and temperature is given by \(\frac{P_1}{T_1} = \frac{P_2}{T_2}\). This formula is derived from the ideal gas law when volume does not change: \(P = nRT/V\).

Rearrange the Formula to Solve for \(P_2\)

Rearrange the formula \(\frac{P_1}{T_1} = \frac{P_2}{T_2}\) to solve for \(P_2\). This gives the formula \(P_2 = P_1 \times \frac{T_2}{T_1}\).

Substitute Known Values

Substitute the known values into the formula: \(P_2 = 50,000 \times \frac{350}{300}\).

Perform the Calculation

Calculate the value of \(P_2\) using the substituted values. \(P_2 = 50,000 \times \frac{350}{300} = 50,000 \times 1.1667 \approx 58,333.33\,\mathrm{Pa}\).

Review the Solution

Review the calculations to ensure accuracy and verify that the derived pressure \(P_2 = 58,333.33\,\mathrm{Pa}\) makes sense, given that temperature increased.

Key concepts

Pressure-Temperature Relationship

The pressure-temperature relationship in gases is a fundamental concept in thermodynamics. This relationship is part of the ideal gas law, which describes how gases behave under various conditions. When the volume of a gas is kept constant, any change in the gas's temperature will directly affect its pressure.
For a fixed amount of gas, an increase in temperature results in an increase in pressure. This is because the molecules of the gas move faster and collide with the walls of the container more frequently and with greater force. As a result, the pressure inside the container increases. Conversely, if the temperature decreases, the pressure decreases as well.
In mathematical terms, this relationship is often expressed using the formula:\[\frac{P_1}{T_1} = \frac{P_2}{T_2}\]
where:

• \(P_1\) and \(P_2\) are the initial and final pressures.
• \(T_1\) and \(T_2\) are the initial and final temperatures, measured in Kelvin.

The formula indicates that the ratio of pressure to temperature remains constant if the volume does not change.

Fixed Volume Gas Behavior

In scenarios where the volume of a gas is held constant, such as in a sealed container, the behavior of the gas predominantly depends on pressure and temperature changes. This is particularly modeled by the isochoric process in thermodynamics where volume does not vary. In such a case, the amount of gas and the physical constraints of the container remain unchanged.
When the gas is heated, its particles gain energy and start moving more vigorously. Since the particles cannot expand within the confines of the fixed volume, the increased movement leads to a rise in pressure. This is seen in the example exercise where the temperature of argon gas is increased, resulting in a higher pressure within the container.
Understanding the behavior of gases in a fixed volume is crucial in various applications such as pressure cookers, internal combustion engines, and even in predicting weather changes in enclosed environments.

Direct Proportionality of Pressure and Temperature

For an ideal gas at constant volume, there is a direct proportionality between pressure and temperature. This means that as one variable increases, the other does the same, assuming pressure is measured in absolute units (such as Pascals) and temperature is in Kelvin. This proportionality is a direct consequence of the kinetic theory of gases, which states that the energy of gas particles relates directly to temperature.
In the given exercise, we see this principle in action. By increasing the temperature of argon gas from 300 K to 350 K, the pressure also rises proportionally from 50,000 Pa to approximately 58,333 Pa. The formula \(P_2 = P_1 \times \frac{T_2}{T_1}\) clearly demonstrates this linear relationship between temperature and pressure.
This proportionality helps to understand and predict the behavior of gases in confined spaces, and it's used in designing safety standards for containers and devices that operate under changing temperature conditions. Knowing how gas pressure will change with temperature ensures safe and efficient operation in industrial and everyday applications.