Question

Suppose a 24.3 -mL sample of helium gas at 25 and 1.01 atm is heated to \(50 .\) C and compressed to a volume of \(15.2 \mathrm{~mL}\). What will be the pressure of the sample?

Short answer

The final pressure of the helium gas sample, after being heated to \(50°C\) and compressed to a volume of \(15.2 mL\), is approximately \(1.76 atm\).

Step-by-step solution

Convert temperatures to Kelvins

We need to convert the Celsius temperatures to Kelvin for our calculations. The formula for converting Celsius to Kelvin is K = °C + 273.15. \( T1 = 25°C + 273.15 = 298.15K \) \( T2 = 50°C + 273.15 = 323.15K \)

Apply the combined gas law formula

We will use the combined gas law formula: \( P1 × V1 / T1 = P2 × V2 / T2 \) Plug in the given values and solve for P2: \( P2 = (P1 × V1 × T2) / (V2 × T1) \)

Calculate the final pressure P2

Use the given values and the temperatures we converted to Kelvin to find the final pressure: \( P2 = (1.01 atm × 24.3 mL × 323.15 K) / (15.2 mL × 298.15 K) \)

Solve for P2

Perform the calculations: \( P2 ≈ (1.01 × 24.3 × 323.15) / (15.2 × 298.15) \) \( P2 ≈ 7955.88145 / 4531.438 \) \( P2 ≈ 1.76 atm \) The final pressure of the helium gas sample will be approximately 1.76 atm.

Key concepts

Understanding Gas Laws

The behavior of gases can be described by several laws, known collectively as the gas laws. These laws explain how the temperature, volume, and pressure of a gas are interrelated. When gas conditions change, as long as the amount of gas and the amount of molecules remains constant, the relationship between these variables is predictable and can be expressed mathematically. Among the most important gas laws are Boyle's Law, Charles's Law, Gay-Lussac's Law, and Avogadro's Law. However, the combined gas law incorporates elements from the first three to provide a more comprehensive equation.

The combined gas law is particularly useful as it allows us to predict the state of a gas sample (pressure, volume, and temperature) after any changes, assuming no gas is added or removed from the system. This law can be expressed as follows:
\[ \frac{P1 \times V1}{T1} = \frac{P2 \times V2}{T2} \]
where:\[ P = \text{pressure} \], \[ V = \text{volume} \], and \[ T = \text{temperature in Kelvin} \]. It is essential to use Kelvin for temperatures because it is an absolute scale, meaning 0 K represents the absence of thermal energy.

Temperature Conversion in Gas Laws

The temperature conversion from Celsius to Kelvin is a critical step in applying the gas laws accurately. This is because gas laws require temperature to be measured on an absolute scale. The Kelvin scale is used for this purpose, as it begins at absolute zero, which is theoretically the lowest possible temperature where all kinetic motion ceases.

To convert Celsius to Kelvin, we use the formula:\[ K = {}^\circ C + 273.15 \]
This step is essential when utilizing the combined gas law, as incorrect temperature units can lead to inaccurate results. For students to accurately solve problems involving gas laws, remembering to convert the temperature to Kelvin, as this will yield a directly proportional relationship between all variables within the law, ensuring that all parts of the equation are consistent.

Pressure-Volume Relationship in Gases

The pressure-volume relationship is one of the fundamental aspects of gas behavior. As illustrated by Boyle's Law, the pressure of a gas sample is inversely proportional to its volume when temperature is held constant:\[ P \times V = \text{constant} \]
Thus, if volume decreases, pressure increases, assuming the amount of gas and temperature remain unchanged. This inverse relationship is observable in everyday life, for example, when you use a bicycle pump to compress air inside the tire—the volume decreases, and the pressure increases.

In the context of the combined gas law, the pressure-volume relationship is intertwined with temperature, signaling that when considering changes in gas conditions, it is vital to account for the three variables collectively. Solving problems involving the pressure-volume relationship requires careful manipulation of the combined gas law formula to isolate and compute the unknown variable, utilizing correct units and temperature conversion.