Problem 27 - Chemistry The Central Science In Si Units

Question

(a) Amonton's law expresses the relationship between pressure and temperature. Use Charles's law and Boyle's law to derive the proportionality relationship between \(P\) and \(T .(\mathbf{b})\) If a car tire is filled to a pressure of 220.6 kPa measured at \(24^{\circ} \mathrm{C}\), what will be the tire pressure if the tires heat up to \(49^{\circ} \mathrm{C}\) during driving?

Step-by-Step Solution

Answer

The tire pressure increases to approximately 239.2 kPa when heated to 49°C.

Step 1: Understand the Laws Involved

Charles's Law states that the volume of a gas is directly proportional to its temperature when pressure is constant: \( V \propto T \). Boyle's Law states that the volume of a gas is inversely proportional to its pressure when temperature is constant: \( V \propto \frac{1}{P} \). These laws describe the behaviors of gases under certain conditions.

Step 2: Combine Charles's and Boyle's Laws

By combining Charles's Law: \( V \propto T \) and Boyle's Law: \( V \propto \frac{1}{P} \), we find that \( T \propto \frac{1}{V} \times P \), leading to \( PV \propto T \). This means that the product of pressure and volume is proportional to the temperature of the gas when the amount of gas is constant. For Amonton's Law, where volume is constant, we can simplify this to \( P \propto T \), indicating pressure and temperature are directly proportional when volume is constant.

Step 3: Convert Temperatures to Kelvin

To apply the gas laws, temperatures must be in Kelvin. Convert the initial temperature: \( 24^{\circ} \mathrm{C} + 273.15 = 297.15\, \mathrm{K}\). Convert the final temperature: \( 49^{\circ} \mathrm{C} + 273.15 = 322.15\, \mathrm{K}\).

Step 4: Use Amonton's Law to Calculate Final Pressure

Using the relationship \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \), we can solve for the final pressure \( P_2 \). Plug in the given values: \( \frac{220.6 \, \mathrm{kPa}}{297.15 \, \mathrm{K}} = \frac{P_2}{322.15 \, \mathrm{K}} \). Solving for \( P_2 \) gives: \( P_2 = \frac{220.6 \, \mathrm{kPa} \times 322.15 \, \mathrm{K}}{297.15 \, \mathrm{K}} \approx 239.2 \, \mathrm{kPa} \).

Key Concepts

Amonton's Law

Amonton's Law, also known as Gay-Lussac's Law, describes the direct relationship between the pressure of a gas and its temperature when the volume is held constant. If the temperature of a gas increases, the pressure also increases, provided the gas is confined in a fixed volume. This is because the molecules are moving faster at higher temperatures, leading to more frequent impacts against the walls of the container, thereby increasing pressure. Mathematically, it is expressed as \( P \propto T \), or equivalently, \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \), where \( P \) is the pressure, \( T \) is the temperature in Kelvin, and the subscripts 1 and 2 refer to the initial and final states of the gas respectively. Understanding this relationship is essential for practical applications, such as calculating the change in tire pressure as temperature changes.

Charles's Law

Charles's Law defines the direct proportionality between the volume of a gas and its temperature, assuming the pressure remains constant. This means that as the temperature of a gas increases, its volume increases, and if the temperature decreases, the volume reduces as well. The law can be mathematically expressed as \( V \propto T \), or \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \). This relationship helps in understanding how gases expand when heated and contract when cooled. To use Charles's Law, it's crucial to ensure temperatures are measured in Kelvin to accurately predict changes in volume under set pressure conditions.

Boyle's Law

Boyle's Law explains how the volume of a fixed quantity of gas is inversely proportional to its pressure, provided the temperature is constant. This can be expressed through the equation \( P \propto \frac{1}{V} \), or \( P_1 V_1 = P_2 V_2 \). When you compress a gas by decreasing its volume, its pressure increases. Conversely, expanding the gas decreases its pressure. Boyle's Law is fundamental in understanding how gases behave under varying pressures and is particularly useful for calculations in closed systems where temperature does not change.

Temperature Conversion

Temperature conversion is critical for applying gas laws since they rely on absolute temperature measurements. In scientific contexts, converting Celsius to Kelvin is a fundamental step because the Kelvin scale is the SI unit of temperature. The conversion is simple: add 273.15 to the Celsius temperature to obtain the temperature in Kelvin. For example, to convert 24°C to Kelvin, compute as follows: \( 24^{\circ}C + 273.15 = 297.15 \, K \). Similarly, to convert 49°C to Kelvin, use \( 49^{\circ}C + 273.15 = 322.15 \, K \). This conversion ensures accuracy when using formulas dependent on proportional relationships between gas variables.

Kelvin Scale

The Kelvin scale is an absolute temperature scale starting at absolute zero, the point at which molecular motion theoretically ceases. Unlike Celsius or Fahrenheit, the Kelvin scale is based on absolute values, making it extremely useful in scientific calculations such as those involving gas laws. Zero Kelvin is \( -273.15^{\circ}C \), and this scale avoids negative temperatures by defining it's starting point at absolute zero. All gas laws require temperature inputs in Kelvin, because the relationships between pressure, volume, and temperature are based on absolute temperature. Understanding how to utilize the Kelvin scale is essential for correctly applying gas law equations and achieving precise results.