Problem 46 - Chemistry for Allied Health

Question

A sample of nitrogen is at \(45^{\circ} \mathrm{C}\) with a volume of \(2.5 \mathrm{~L}\). What is the final temperature in \({ }^{\circ} \mathrm{C}\) if the volume is compressed to \(1.4\) L? Assume constant pressure and moles.

Step-by-Step Solution

Answer

The final temperature is approximately \(-94.81^{\circ} \text{C}\).

Step 1: Identify the Known Variables

We know the initial volume of the nitrogen gas, \(V_1 = 2.5 \text{ L}\), and the final volume, \(V_2 = 1.4 \text{ L}\). The initial temperature is given in Celsius, \(T_1 = 45^{\circ} \text{C}\). To use the gas laws, we should convert this temperature to Kelvin using the formula: \(T(K) = T(^{\circ}C) + 273.15\). Thus, \(T_1 = 45 + 273.15 = 318.15 \text{ K}\).

Step 2: Apply Charles's Law

Since the pressure and number of moles are constant, Charles's Law applies, stating that the volume of a gas is directly proportional to its temperature in Kelvin. The formula for Charles's Law is: \[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \] where \(T_2\) is the unknown final temperature in Kelvin.

Step 3: Solve for the Final Temperature

Rearrange the formula to solve for \(T_2\): \[ T_2 = \frac{V_2 \times T_1}{V_1} \] Substitute the values: \[ T_2 = \frac{1.4 \text{ L} \times 318.15 \text{ K}}{2.5 \text{ L}} \approx 178.344 \text{ K} \]

Step 4: Convert Final Temperature Back to Celsius

Convert the temperature from Kelvin back to Celsius using the formula: \(T(^{\circ}C) = T(K) - 273.15\). Thus, \(T_2 = 178.344 - 273.15 \approx -94.81^{\circ} \text{C}\).

Key Concepts

Gas Laws

Gas laws are fascinating principles that describe how different variables affect the behavior of gases. They include laws like Boyle's Law, Charles's Law, and Avogadro's Law, each explaining different relationships between pressure, volume, temperature, and the amount of gas. Here, our focus is on Charles's Law. Charles's Law involves the volume and temperature of a gas at constant pressure and amount of gas. This law states that the volume of a gas is directly proportional to its absolute temperature (in Kelvin) when pressure is held constant. In simple terms, if you increase the temperature of the gas, its volume will also increase and vice versa. Understanding these laws helps predict how a gas will react under different conditions, which is crucial in various scientific and industrial applications.

Temperature Conversion

Temperature conversion is a crucial step in calculating and understanding gas behaviors. Temperature is often given in degrees Celsius, but for gas law calculations, it must be converted to Kelvin. This is because the Kelvin scale is an absolute scale, starting at zero, which represents absolute zero—the point at which molecular motion stops. Converting Celsius to Kelvin is straightforward:

• Add 273.15 to the Celsius temperature to convert to Kelvin.
• This conversion ensures that the temperature scale used does not have negative values, which could complicate proportional relationships.

For example, if the temperature is 45°C, you add 273.15 to get 318.15 K, ensuring accurate calculations when using gas laws. Remember, it's always essential to convert back to Celsius if the problem asks for the final answer in Celsius.

Volume and Temperature Relationship

The volume and temperature relationship in gases is elegantly explained by Charles's Law, which states that the two quantities are directly proportional, provided pressure remains constant. When you increase the temperature of a gas, its molecules gain kinetic energy and move faster, causing the gas to expand and the volume to increase. Conversely, cooling the gas causes it to contract. In practical terms, this means:

• If a gas's temperature is doubled, its volume also doubles, assuming pressure and amount of gas remain constant.
• Compressing a gas (reducing its volume) while keeping the same amount of gas and pressure requires lowering the temperature.

In our scenario, as the gas volume decreased from 2.5 L to 1.4 L, the temperature had to decrease as well, evident from our solution, ultimately finding the final temperature in Celsius.