Problem 36 - A Complete Resource Book in Chemistry for JEE Main

Question

If two moles of ideal gas at \(540 \mathrm{~K}\) has volume \(44.8 \mathrm{~L}\), then its pressure will be (a) \(1 \mathrm{~atm}\) (b) \(2 \mathrm{~atm}\) (c) \(3 \mathrm{~atm}\) (d) \(4 \mathrm{~atm}\)

Step-by-Step Solution

Answer

The pressure is approximately \( 2 \) atm, so the answer is (b) \( 2 \) atm.

Step 1: Identify the Ideal Gas Law

The ideal gas law is given by the equation \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is temperature. Our goal is to find pressure \( P \).

Step 2: Assign Known Values

Based on the information given, assign the values: \( n = 2 \) moles, \( V = 44.8 \) L, \( T = 540 \) K. The universal gas constant \( R \) can be either \( 0.0821 \) L atm / (K mol) when using atm as the pressure unit.

Step 3: Plug Values into Ideal Gas Law

Substitute the known values into the formula: \( P = \frac{nRT}{V} = \frac{2 \times 0.0821 \times 540}{44.8} \).

Step 4: Calculate the Pressure

Calculate the expression: \( P = \frac{2 \times 0.0821 \times 540}{44.8} = \frac{88.668}{44.8} \approx 1.98 \). The pressure is approximately \( 2 \) atm.

Key Concepts

Pressure Calculation

Calculating the pressure of a gas is a fundamental step in understanding the behavior of gases under different conditions. This is particularly useful in real-world applications such as meteorology and engineering. The pressure of a gas is influenced by various factors, including the volume it occupies, the number of moles of gas present, and its temperature.

• Volume: As the volume decreases, keeping everything else constant, the pressure increases. This is known as Boyle's Law.
• Temperature: Increasing temperature generally increases pressure, assuming the volume is constant, which aligns with Charles's Law.
• Moles of Gas: More moles of gas at constant volume and temperature mean higher pressure, showcasing Avogadro's principle.

To calculate pressure using the ideal gas law formula is quite straightforward. By rearranging the equation to solve for pressure, you'll use the formula: \( P = \frac{nRT}{V} \) This allows us to find the pressure when the number of moles, temperature, and gas constant are known.

Universal Gas Constant

The universal gas constant, \( R \), is a vital component in the ideal gas law equation. It acts as a bridge between the measurable properties of gas, linking pressure, volume, temperature, and the number of moles into one cohesive equation. In our calculations, 'R' is given as \(0.0821 \ \text{L atm} / (\text{K mol})\).

• This value of 'R' is used when pressure is measured in atmospheres and volume in liters.
• The constant 'R' must be consistent with the units of the other variables in the equation; otherwise, the calculations will not yield an accurate result.

Understanding the role of 'R' helps align calculations with actual physical conditions, ensuring precision and accuracy in predicting how gases will behave under specified conditions.

Moles of Gas

The concept of moles is central to chemistry, representing an amount of substance. In the context of gases, moles allow us to quantify the particles present in a given volume.

• One mole is equivalent to Avogadro's number, approximately \(6.022 \times 10^{23}\) particles.
• In gas calculations, the number of moles directly influences the total amount of gas particles available, affecting both pressure and volume.

For example, in our exercise, we consider two moles of gas, a significant quantity affecting the overall system's pressure when placed in a confined volume. By knowing the moles, we can predict and calculate changes in other properties of the gas when manipulated under different conditions, such as changes in temperature or volume.