Question

One litre of a gas collected at NTP will occupy ..... litre at 2 atm pressure and \(27^{\circ} \mathrm{C}\) (1) \(\frac{300}{2 \times 273}\) litre (2) \(\frac{2 \times 300}{273}\) litre (3) \(\frac{273}{2 \times 300}\) litre (4) \(\frac{2 \times 273}{300}\) litre

Short answer

Option 1: \( \frac{300}{2 \times 273} \) litre

Step-by-step solution

Recall Ideal Gas Law

The ideal gas law is given by\[PV = nRT\]. This will help to understand the relations between pressure, volume, and temperature.

Understand NTP conditions

At NTP (Normal Temperature and Pressure), the conditions are: Pressure \(P_1 = 1\) atm, Temperature \(T_1 = 273\) K, and Volume \(V_1 = 1\) litre.

Note the given conditions

The gas is now at a pressure \(P_2 = 2\) atm and temperature \(T_2 = 27^{\text{o}} \text{C} = 300\text{K}\). We need to find the new volume \(V_2\)

Apply combined gas law

Use the combined gas law which relates initial and final states of a gas:\[\frac{P_1}{T_1} \frac{V_1}{n} = \frac{P_2}{T_2} \frac{V_2}{n} \]. Since the amount of gas \(n\) remains constant, the formula simplifies to \(\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\).

Insert known values

Insert the known values into the simplified equation:\[\frac{1 \times 1}{273} = \frac{2 \times V_2}{300}\].

Solve for the new volume

Solve for \(V_2\) to find the new volume:\[V_2 = \frac{300}{2 \times 273} = \frac{300}{546} \text{ litres}\].

Key concepts

Ideal Gas Law

The Ideal Gas Law is crucial for understanding how gases behave under different conditions. It's represented by the equation \(PV = nRT\), where:

• **P** is the pressure of the gas.
• **V** is the volume.
• **n** is the number of moles of gas.
• **R** is the gas constant.
• **T** is the temperature in Kelvin.

This equation helps us understand the relationship between these variables. For instance, if we have a certain amount of gas at a known temperature and pressure, we can determine its volume. In the given exercise, using this law allows us to predict the volume change when pressure and temperature conditions change. Remember, the moles of gas (n) and the gas constant (R) remain constant in most scenarios.

Combined Gas Law

The Combined Gas Law merges Boyle's Law, Charles's Law, and Gay-Lussac's Law to relate the pressure, volume, and temperature of a gas. It's expressed as \(\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\). This formula is useful in solving the given problem because it allows us to compare two different sets of pressure, volume, and temperature conditions. Here's how to use it:

• Start by identifying your initial and final conditions (\(P_1, V_1, T_1\) and \(P_2, V_2, T_2\)).
• Plug these values into the equation and solve for the unknown variable, which in our case is the final volume \(V_2\).

By understanding and applying the combined gas law, you can figure out how changes in conditions will affect the volume of the gas.

Normal Temperature and Pressure (NTP)

Normal Temperature and Pressure, or NTP, is a standard set of conditions used in various scientific calculations. At NTP, the conditions are defined as:

• Pressure, \(P_1 = 1 \text{ atm}\).
• Temperature, \(T_1 = 273 \text{ K}\).
• Often, the volume of one mole of an ideal gas is considered to be 22.4 liters.

These conditions offer a baseline that scientists use to compare different experiments. For our exercise, NTP conditions are used as the starting point to understand how the gas behaves when the pressure is increased to 2 atm and the temperature to 300 K. By starting from a standard condition, it’s easier to predict how changes will affect the gas volume.

Volume and Pressure Relationship

The relationship between volume and pressure in gases is described by Boyle's Law. This law states that the volume of a gas is inversely proportional to its pressure when temperature is constant, written as \(P_1 V_1 = P_2 V_2\). This means:

• If the pressure increases, the volume will decrease, provided the temperature remains the same.
• Similarly, if the pressure decreases, the volume will increase.

In our problem, we need to consider both the pressure change and the temperature change. By combining Boyle’s Law with Charles’s Law (which covers temperature) in the form of the Combined Gas Law, we can accurately predict the new volume of the gas. Understanding these relationships is key to solving problems involving gas behaviors.