Question

The temperature of 20 litres of nitrogen was increased from \(100 \mathrm{~K}\) to \(300 \mathrm{~K}\) at constant pressure. Change in volume will be (1) 80 litres (2) 60 litres (3) 40 litres (4) 20 litres

Short answer

The change in volume is 40 liters.

Step-by-step solution

Understand the Problem

The problem involves the relationship between temperature and volume of nitrogen gas at a constant pressure. The temperature increases from 100 K to 300 K. Given the initial volume is 20 liters, we need to find the new volume and calculate the change.

Use Charles's Law

Charles's Law states that at constant pressure, the volume of a gas is directly proportional to its temperature (in Kelvin). The formula is \( V_1 / T_1 = V_2 / T_2 \). Here, \( V_1 = 20 \text{ liters} \), \( T_1 = 100 \text{ K} \), and \( T_2 = 300 \text{ K} \).

Rearrange the Formula

To find the final volume \( V_2 \), rearrange the formula: \( V_2 = V_1 \times (T_2 / T_1) \).

Substitute the Values

Substitute \( V_1 = 20 \text{ liters} \), \( T_1 = 100 \text{ K} \), and \( T_2 = 300 \text{ K} \) into the formula: \( V_2 = 20 \times (300 / 100) = 20 \times 3 = 60 \text{ liters} \).

Calculate the Change in Volume

The change in volume is \( \text{Final Volume} - \text{Initial Volume} = 60 \text{ liters} - 20 \text{ liters} = 40 \text{ liters} \).

Key concepts

Charles's Law

Charles's Law is a fundamental principle in the study of gases. It describes how gases tend to expand when heated. The law states that at constant pressure, the volume of a gas is directly proportional to its temperature, provided the temperature is measured in Kelvin.
The formula for Charles's Law is: \[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]
Here,

• \(V_1\) represents the initial volume of the gas,
• \(T_1\) represents the initial temperature,
• \(V_2\) represents the final volume of the gas, and
• \(T_2\) represents the final temperature.

In simple terms, this law means that if you increase the temperature of a gas, its volume will also increase as long as the pressure remains constant. This is because the gas molecules move faster and spread out more when they are heated.

Temperature-Volume Relationship

The temperature-volume relationship is an important concept in understanding how gases behave under different conditions. According to Charles's Law, this relationship suggests that the volume of a gas will increase as the temperature increases, assuming the pressure is kept constant.
For instance, if you have 20 liters of nitrogen gas at a temperature of 100 K and you heat it up to 300 K, the volume will increase. The steps to calculate the new volume are:
1. Identify the initial volume (\(V_1\)) and temperatures (\(T_1\) and \(T_2\)).
2. Use the Charles's Law formula to solve for the final volume (\(V_2\)).
Using the formula: \(V_2 = V_1 \times (T_2/T_1)\),
Substituting the known values: \(V_2 = 20 \times (300/100) = 60 \text{ liters} \).
This means that the gas expands to 60 liters when the temperature is increased from 100 K to 300 K.

Ideal Gas Law

The ideal gas law combines several critical concepts, including the temperature-volume relationship described by Charles's Law. The formula for the ideal gas law is: \[ PV = nRT \] Here,

• \(P\) represents the pressure,
• \(V\) represents the volume,
• \(n\) represents the number of moles of the gas,
• \(R\) is the gas constant, and
• \(T\) represents the temperature in Kelvin.

The ideal gas law connects the principles of pressure, volume, and temperature, providing a comprehensive framework for understanding gas behavior.
While Charles's Law focuses specifically on the temperature-volume relationship at constant pressure, the ideal gas law is broader. It also takes into account changes in pressure and the amount of gas. Understanding the ideal gas law helps in predicting how a gas will respond to various changes in temperature, volume, or pressure.
In the example problem, only Charles's Law is directly applicable because pressure remains constant while temperature and volume change.