Question

What is the pressure of a fixed volume of hydrogen gas at 30.0°C if it has a pressure of 1.11 atm at 15.0°C?

Short answer

The pressure of the fixed volume of hydrogen gas at 30.0°C is approximately \(1.173\, \text{atm}\).

Step-by-step solution

Write down the equation

We will use the combined gas law equation which states: \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \) Where - \(P_1\) is the initial pressure, - \(T_1\) is the initial temperature in Kelvin, - \(P_2\) is the final pressure, and - \(T_2\) is the final temperature in Kelvin.

Convert temperatures to Kelvin

We need to convert the given temperatures from Celsius to Kelvin. The conversion is as follows: \(K = °C + 273.15\) Initial temperature (\(T_1\)) in Kelvin: \(T_1 = 15.0°C + 273.15 = 288.15K\) Final temperature (\(T_2\)) in Kelvin: \(T_2 = 30.0°C + 273.15 = 303.15K\)

Plug the values into the equation

Now we will plug the known values (\(P_1\), \(T_1\), and \(T_2\)) into the combined gas law equation and solve for the final pressure (\(P_2\)). \( \frac{1.11\,\text{atm}}{288.15\,\text{K}} = \frac{P_2}{303.15\,\text{K}} \)

Solve for the final pressure (\(P_2\))

To find the value of \(P_2\), we will rearrange the equation and then perform the calculation: \( P_2 = \frac{1.11\,\text{atm} \times 303.15\,\text{K}}{288.15\,\text{K}} \) \( P_2 = 1.173\,\text{atm} \)

Present the final result

The pressure of the fixed volume of hydrogen gas at 30.0°C is approximately \(1.173\, \text{atm}\).

Key concepts

Gas Law Equations

Gas law equations are the cornerstone of understanding behavior of gases under different conditions. The combined gas law is particularly useful as it combines three fundamental gas laws, which are Charles's Law (volume-temperature relationship), Boyle's Law (pressure-volume relationship), and Gay-Lussac's Law (pressure-temperature relationship).

The combined gas law is represented by the formula: \( P_1 / T_1 = P_2 / T_2 \) when the number of moles and the volume remain constant. This equation shows how the pressure and temperature of a gas are related to each other. When using this equation, the most critical step is to ensure that the temperatures are always expressed in Kelvin, the SI unit for thermodynamic temperature, as Celsius does not reflect an absolute scale where proportions hold true.

Temperature Conversion

Temperature conversion between Celsius and Kelvin is a key aspect when working with gas law equations, as mentioned earlier. Since gas laws require the temperature to be in Kelvin, it is crucial to convert Celsius temperatures before plugging them into the equations.

The formula for converting Celsius to Kelvin is: \( K = \degree C + 273.15 \). This formula is derived from the fact that the Kelvin and Celsius scales have the same magnitude of degree, but different starting points—0 Kelvin is equal to -273.15°C. Therefore, 0°C, the freezing point of water, is equivalent to 273.15K. It's important to perform this conversion carefully to avoid any mistakes in subsequent calculations.

Pressure-Temperature Relationship

The pressure-temperature relationship is described by Gay-Lussac's Law, which states that the pressure exerted by a gas is directly proportional to its temperature when the volume and the amount of gas are held constant. As the temperature of a gas increases, so does its pressure, if the volume doesn't change. The same relationship is part of the combined gas law.

This direct relationship means that as one value increases, the other also increases at a constant rate. This concept is central to figuring out how the pressure and temperature vary in tandem under controlled conditions. When solving problems involving pressure-temperature relationships, it is fundamental to keep in mind the units used for each value, especially ensuring that temperature is converted to Kelvin.