Question

The temperature of \(2.5 \mathrm{~L}\) of a gas initially at STP is increased to \(250^{\circ} \mathrm{C}\) at constant volume. Calculate the final pressure of the gas in atm.

Short answer

Computing the equation above, the final pressure \(\(P2\)\) turns out to be approximately \(1.915 atm\). So, the final pressure of the gas after being heated to \(250^{\circ} \mathrm{C}\) at constant volume is approximately \(1.915 atm\).

Step-by-step solution

Understand the problem and data

You initially have a gas at STP, meaning the gas is at a standard temperature of \(0^{\circ} C\) (or \(273.15 K\)) and a standard pressure of \(1 atm\). The temperature is then raised to \(250^{\circ} C\), while the volume remains constant, and the final pressure is to be found.

Convert temperatures to the Kelvin scale

The formulae in gas laws typically involve absolute temperature, meaning the Kelvin scale must be used. To convert from Celsius to Kelvin, we add \(273.15\) to the Celsius temperature. That's why the initial temperature \(0^{\circ} C\) becomes \(0 + 273.15 = 273.15 K\) and the final temperature \(250^{\circ} C\) becomes \(250 + 273.15 = 523.15 K\).

Apply Gay-Lussac's Law

According to Gay-Lussac's Law, the pressure of a gas is directly proportional to its temperature if its volume is constant. Hence, \(\frac{P1}{T1} = \frac{P2}{T2}\). Given that \(P1 = 1 atm\), \(T1 = 273.15 K\), \(T2 = 523.15 K\), we need to solve for \(P2 = \frac{P1 × T2}{T1}\).

Calculate final pressure

Substituting the known values into the formula, you get \(P2 = \frac{1 atm × 523.15 K}{273.15 K}\). Solving this calculation will provide the final pressure value.

Key concepts

Gay-Lussac's Law

Gay-Lussac's Law explains how the pressure of a gas changes in relation to its temperature when the volume remains constant. This law is an important part of understanding gas behaviors. The relationship it describes is direct: as the temperature of a gas rises, so does the pressure if the volume does not change.
This principle can be represented mathematically by the equation: \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \). Here, \( P_1 \) and \( T_1 \) represent the initial pressure and temperature, while \( P_2 \) and \( T_2 \) denote the final pressure and temperature, respectively.
Using Gay-Lussac's Law requires converting temperatures to the Kelvin scale to ensure accuracy. In practical scenarios, like the exercise provided, you can see how increasing the temperature at a constant volume will increase the pressure, as calculated through this law. This understanding is essential in many scientific and engineering applications.

Kelvin scale

The Kelvin scale is crucial in thermodynamics and gas laws because it is an absolute temperature scale. This means it starts from absolute zero, the theoretical point where particles have minimal thermal motion.
Unlike Celsius or Fahrenheit, Kelvin does not use degrees, just kelvins (K). To convert from Celsius to Kelvin, simply add 273.15 to the Celsius temperature.

• Example: Convert \(250^\circ C\) to Kelvin: \(250 + 273.15 = 523.15\) K.

Using the Kelvin scale is important for ensuring calculations adhere to scientific standards. For gas laws like Gay-Lussac's, where temperature is involved, converting to Kelvin ensures that the proportional relationships between variables remain consistent and correct.

STP (Standard Temperature and Pressure)

STP stands for Standard Temperature and Pressure, a commonly used reference point for studying gas behaviors. At STP, the conditions are defined as a temperature of \(0^\circ C\) (or 273.15 K) and a pressure of 1 atmosphere (atm).
Using STP makes comparing different experiments and calculations straightforward, as it provides a uniform basis. The exercise illustrates how changes in temperature affect gas pressure starting from STP conditions, a typical scenario in many textbooks and real-world situations involving gases.

• At STP, many gases behave in a predictable manner, as described by various gas laws.
• These conditions help scientists and engineers ensure results are repeatable and consistent across different experiments.

Understanding STP is fundamental for anyone studying chemistry, physics, or engineering, as it sets the stage for measuring and predicting gas reactions and behaviors.