Problem 7 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

At \(0^{\circ} \mathrm{C}\) and a pressure of \(1000 \mathrm{~mm}\), a given weight of nitrogen occupies a volume of \(1.0\) 1. At \(-100^{\circ} \mathrm{C}\), the same weight of gas under the same pressure occupies a volume of \(0.61\). What is the value of absolute zero in degree celsius? (a) \(-250^{\circ} \mathrm{C}\) (b) \(250^{\circ} \mathrm{C}\) (c) \(-273^{\circ} \mathrm{C}\) (d) \(273^{\circ} \mathrm{C}\)

Step-by-Step Solution

Answer

-273^{\text{o}} C

Step 1: Understand Charles's Law

Charles's Law states that, at constant pressure, the volume of a given mass of an ideal gas is directly proportional to its temperature measured in Kelvin. It can be expressed as V1/T1 = V2/T2 where V1 and V2 are volumes and T1 and T2 are temperatures in Kelvin.

Step 2: Convert temperatures to Kelvin

To apply Charles's Law, the temperatures must be in Kelvin. Convert the given temperatures from Celsius to Kelvin by adding 273 to each. Thus, T1 is 0 + 273 = 273 K and T2 is -100 + 273 = 173 K.

Step 3: Plug values into Charles's Law

The volumes are given as V1 = 1.0 L at 273 K and V2 = 0.61 L at 173 K. We can now set up the proportion 1.0/273 = 0.61/173.

Step 4: Solve for the absolute zero in Celsius

To find the value of absolute zero, we consider what happens as V approaches 0. For V to be 0, the temperature would also have to be 0 K. Setting up the proportion with V being 0, T (in Kelvin) should also be 0. Using the proportion from Step 3, if V=0, we solve for T in Kelvin and then convert it back to Celsius by subtracting 273 to find the value of absolute zero.

Step 5: Determine the proportion to predict absolute zero

Using the proportion from Step 3, we can create an equation. If V2 = (V1 * T2) / T1, and we know that at absolute zero V2 will be 0, solve for T1 in terms of T2: 0 = (1.0 * 173) / T1. This implies that T1 = 0 K, which when converted to Celsius gives us -273 degrees Celsius.

Key Concepts

Charles's Law

Charles's Law defines the relationship between the volume and temperature of a given amount of gas at constant pressure. It's a fundamental principle in understanding how gases behave under different temperatures and is essential for calculating changes in gas volumes.

The law can be stated mathematically as: \( V_1/T_1 = V_2/T_2 \), where \( V_1 \) and \( V_2 \) are the initial and final volumes of the gas, and \( T_1 \) and \( T_2 \) are the initial and final temperatures in Kelvin, respectively. The direct proportionality indicates that as the temperature of a gas increases, so does its volume, provided the pressure is held constant.

In a practical sense, this means that if you heat a balloon, it will expand as the gas particles inside move more rapidly, requiring more space. Conversely, cooling a gas will cause it to contract, as the particles slow down and require less space. Charles's Law is particularly relevant when considering the behavior of ideal gases—hypothetical gases that perfectly follow the gas laws. However, this law can still be used to make accurate predictions for real gases under many conditions.

Ideal Gas Volume-Temperature Relationship

The volume-temperature relationship of an ideal gas is closely described by Charles's Law, but it's important to understand the broader context of this behavior. This relationship assumes the gas behaves ideally, which means the gas particles do not attract or repel each other, and they have negligible volume compared to the space the gas occupies.

Ideal gases are a simplified model used to make predictions about gas behavior under various conditions. These predictions become less accurate under extreme conditions, such as high pressure or very low temperature, where real gases deviate from ideal behavior. Nonetheless, the ideal gas model is quite useful for learning and applying gas laws.

The relationship between the volume and temperature of an ideal gas is critical for various applications, from designing engines and refrigerators to understanding atmospheric science. When dealing with gas volume and temperature calculations, remember that temperature must be in Kelvin, a scale designed to work seamlessly with the gas laws.

Kelvin Temperature Conversion

The Kelvin is the base unit of temperature in the International System of Units (SI), and it's fundamental to the study of thermodynamics and ideal gas laws. Unlike Celsius or Fahrenheit, Kelvin starts at absolute zero, which is the theoretical point at which all molecular motion stops.

Converting Celsius to Kelvin is relatively straightforward; simply add 273.15 to your Celsius temperature: \( T(K) = T(°C) + 273.15 \). This conversion places the freezing point of water at 273.15 K and the boiling point at 373.15 K.

Understanding how to convert temperatures to Kelvin is crucial when applying Charles's Law, as it ensures that any volume changes due to temperature fluctuations are accurately quantified. This conversion also helps in determining the absolute temperature, which must be used for any calculations involving the ideal gas law or its specific case, Charles's Law.

Seeing temperature in Kelvin allows scientists and engineers to calculate thermal energy, study heat transfer, and explore the universe, from the coldest interstellar clouds to the scorching surfaces of stars.