Problem 59 - Physics

Question

The temperature of an ideal gas is doubled while the volume is kept constant. Does the absolute pressure of the gas double when the temperature that doubles is (a) the Kelvin temperature and (b) the Celsius temperature? Explain. Problem Determine the ratio \(P_{2} / P_{1}\) of the final pressure \(P_{2}\) to the initial pressure \(P_{1}\) when the temperature rises (a) from 35.0 to \(70.0 \mathrm{~K}\) and \((\mathrm{b})\) from 35.0 to \(70.0{ }^{\circ} \mathrm{C}\). Check to see that your answers are consistent with your answers to the Concept Questions.

Step-by-Step Solution

Answer

Pressure doubles with Kelvin; it doesn't with Celsius.

Step 1: Understanding the Problem

We need to determine how the absolute pressure of an ideal gas changes when the temperature changes, keeping the volume constant. We will use the ideal gas law, which relates pressure, volume, and temperature.

Step 2: Using the Ideal Gas Law

For an ideal gas at constant volume, the relationship between pressure and temperature is given by \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \), where \( P \) is pressure and \( T \) is temperature in Kelvin.

Step 3: Converting Temperatures to Kelvin (if needed)

Let's convert Celsius to Kelvin if necessary: \( T(K) = T(^{\circ}C) + 273.15 \).

Step 4: Case (a): Temperature in Kelvin

Given temperatures are from 35.0 K to 70.0 K. Here, \( T_1 = 35.0 \) K and \( T_2 = 70.0 \) K. Plug these into the equation: \( \frac{P_2}{P_1} = \frac{T_2}{T_1} = \frac{70.0}{35.0} = 2 \). The pressure doubles.

Step 5: Case (b): Temperature in Celsius

Given temperatures are from 35.0 °C to 70.0 °C. First, convert these to Kelvin: \( T_1 = 35.0 + 273.15 = 308.15 \) K and \( T_2 = 70.0 + 273.15 = 343.15 \) K. Calculate the pressure ratio: \( \frac{P_2}{P_1} = \frac{T_2}{T_1} = \frac{343.15}{308.15} \approx 1.11 \). The pressure does not double.

Key Concepts

Absolute Pressure

Absolute pressure is the total pressure exerted by a gas, including the atmospheric pressure. It is an essential concept in understanding gas behavior, especially when using the ideal gas law. Absolute pressure is measured in units like pascals (Pa) or atmospheres (atm). It is termed "absolute" to distinguish it from gauge pressure, which only measures the pressure above atmospheric pressure.
In the context of an ideal gas, absolute pressure is directly related to temperature and volume, as described by the ideal gas law. Hence, any change in temperature will affect the absolute pressure of the gas, provided the volume remains constant. When we double the Kelvin temperature of an ideal gas, the absolute pressure also doubles, following the equation \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \). This relationship is crucial in predicting how gas behaves under varying conditions.

Temperature Conversion

Temperature conversion is vital when dealing with gases, as calculations often require temperatures in specific units such as Kelvin. The conversion between Celsius and Kelvin is straightforward: to convert Celsius to Kelvin, simply add 273.15 to the Celsius temperature. Thus, \( T(K) = T(°C) + 273.15 \).
Understanding this conversion is crucial because many gas laws, including the ideal gas law, require temperatures to be in Kelvin for accurate calculations. Whenever a problem involving gas laws uses Celsius as the temperature unit, remember to convert it to Kelvin first. This ensures that calculations remain consistent and yield accurate results.

Kelvin Scale

The Kelvin scale is an absolute temperature scale used in science, including physics and chemistry, to describe temperatures without the influence of the freezing or boiling points of water. The Kelvin scale starts at absolute zero, the lowest possible temperature, where all molecular motion stops. At absolute zero, the Kelvin temperature is zero, which is equivalent to -273.15°C.
The Kelvin scale is particularly important in equations such as the ideal gas law because it ensures that all temperature values are positive and directly proportional to absolute pressure. This proportionality simplifies the relationship between pressure, volume, and temperature, allowing straightforward calculations without the need to adjust for arbitrary zero points like those on the Celsius scale.

Celsius Scale

The Celsius scale is the most commonly used temperature scale for everyday applications, as it aligns well with human experience, with 0°C set at the freezing point of water and 100°C at its boiling point. However, when dealing with scientific calculations, like those involving the ideal gas law, the Celsius scale is not ideal because it does not start at absolute zero.
For calculations involving pressure and temperature, the Celsius scale can be misleading since it can yield negative values, which can complicate proportional relationships inherent in gas laws. Therefore, converting temperatures from Celsius to Kelvin when solving physics or chemistry problems is essential. The Kelvin scale's zero point corresponds to absolute zero, making it a more reliable choice for scientific equations.