Question

Indicate whether the final volume in each of the following is the same, larger, or smaller than the initial volume, if pressure and amount of gas do not change: a. A volume of \(505 \mathrm{~mL}\) of air on a cold winter day at \(-15^{\circ} \mathrm{C}\) is breathed into the lungs, where body temperature is \(37^{\circ} \mathrm{C}\). b. The heater used to heat the air in a hot-air balloon is turned off. c. A balloon filled with helium at the amusement park is left in a car on a hot day.

Short answer

a. larger, b. smaller, c. larger.

Step-by-step solution

Understand the problem

The exercise requires determining the change in the volume of a gas when temperature changes, while pressure and the amount of gas remain constant. This implies the use of Charles's Law, which states that, for a given mass of gas at constant pressure, the volume is directly proportional to its absolute temperature (in Kelvins).

Use Charles's Law formula

According to Charles's Law, \(\frac{V_1}{T_1} = \frac{V_2}{T_2}\). Here, \(V_1\) is the initial volume, \(T_1\) is the initial temperature, \(V_2\) is the final volume, and \(T_2\) is the final temperature.

Convert temperatures to Kelvins

To use Charles's Law, convert the temperatures from Celsius to Kelvins using the formula \(T(K) = T(^{\circ}C) + 273.15\).

Step 4a: Calculate for part (a)

Initial temperature, \(T_1 = -15^{\circ}C + 273.15 = 258.15 \, K\). Final temperature, \(T_2 = 37^{\circ}C + 273.15 = 310.15 \, K\). Since \(T_2 > T_1\), the final volume \(V_2\) will be larger than the initial volume \(V_1\) based on Charles's Law.

Step 4b: Analyze for part (b)

When the heater is turned off, the temperature of the air inside the hot-air balloon decreases. According to Charles's law, if the temperature decreases, the volume will also decrease. Therefore, the volume will be smaller.

Step 4c: Analyze for part (c)

If a balloon filled with helium is left in a car on a hot day, the temperature of the helium increases. According to Charles's Law, an increase in temperature results in an increase in volume. Therefore, the volume will be larger.

Key concepts

Gas Laws

To understand the behavior of gases under different conditions, we use gas laws. These laws describe how gases react to changes in pressure, volume, and temperature. The primary gas laws include Boyle's Law, Charles's Law, and Avogadro's Law.

Charles's Law is particularly important for understanding temperature and volume relationships in gases. It states that, at constant pressure, the volume of a gas is directly proportional to its absolute temperature measured in Kelvin. This means that if the temperature of a gas goes up, its volume also goes up, and if the temperature goes down, then the volume goes down too.

Temperature and Volume Relationship

The relationship between temperature and volume in gases is formally stated by Charles's Law. The formula for Charles's Law is \[\frac{V_1}{T_1} = \frac{V_2}{T_2}\text{,}\] where \( V_1 \) is the initial volume, \( T_1 \) is the initial temperature, \( V_2 \) is the final volume, and \( T_2 \) is the final temperature. Remember, temperatures must always be converted to Kelvin for accurate calculations. The Kelvin scale starts at absolute zero, which is -273.15°C. To convert from Celsius to Kelvin, you add 273.15 to the temperature in degrees Celsius.

In our exercise examples:

• For part (a), the temperature of the air breathed in goes from -15°C (or 258.15 K) to 37°C (or 310.15 K). Since the final temperature is higher, the final volume will be larger.
• For part (b), when the heater in the balloon is turned off, the temperature drops. Consequently, the volume of air inside the balloon also decreases.
• For part (c), leaving a helium balloon in a hot car increases the temperature of the gas. According to Charles's Law, this increase in temperature means the helium will expand, so the volume gets larger.

Constant Pressure Expansion

Constant pressure expansion refers to situations where the volume of a gas changes due to temperature variations while maintaining the same pressure. Under such conditions, Charles's Law precisely describes the behavior of the gas. It's important to understand that temperature and volume changes happen simultaneously and uniformly.

For example, in a hot-air balloon:

• If the air inside is heated (pressure remains the same), it will expand and make the balloon rise.
• Conversely, if the balloon cools down (like when the heater is turned off), the air contracts and the balloon descends.

Both these scenarios depict constant pressure expansion which paints a clear picture of Charles's Law.