Problem 87 - Master Resource Book in JEE Main Physics

Question

If a given mass of gas occupies a volume of \(100 \mathrm{cc}\) at 1 atm pressure and temperature of \(100^{\circ} \mathrm{C}(373.15 \mathrm{~K})\). What will be its volume at 4 atm pressure; the temperature being the same? (a) \(100 \mathrm{cc}\) (b) \(400 \mathrm{cc}\) [c) \(25 \mathrm{cc}\) (d) \(104 \mathrm{cc}\)

Step-by-Step Solution

Answer

The volume is 25 cc (option [c]).

Step 1: Understanding the Initial Conditions

The problem gives us the initial conditions of the gas: its volume is 100 cc, pressure is 1 atm, and temperature is 373.15 K (or 100°C). Our task is to find out the changed volume when the pressure is increased to 4 atm while the temperature remains constant.

Step 2: Applying Boyle's Law

According to Boyle's Law, if the temperature remains constant, the pressure of a gas is inversely proportional to its volume. This can be expressed mathematically as: \[ P_1 V_1 = P_2 V_2 \] Substitute the known values into the equation: \[ 1 \times 100 = 4 \times V_2 \]

Step 3: Solving for the New Volume

Reorganize the equation to solve for \( V_2 \) by dividing both sides by 4: \[ V_2 = \frac{100}{4} \] Calculate to find \( V_2 \): \[ V_2 = 25 \]

Step 4: Conclusion

The volume, when the pressure is 4 atm and the temperature remains the same, is 25 cc. This is option (c).

Key Concepts

Gas Laws

Gas laws are fundamental principles that describe the behavior of gases in relation to pressure, volume, and temperature. Several different gas laws exist, each highlighting a specific relationship between these variables. One of the most critical is Boyle's Law, which focuses on the relationship between pressure and volume when temperature remains constant. Understanding these laws is essential for predicting how a gas will behave under changing conditions. Some commonly known gas laws include:

• Boyle's Law – the relationship between pressure and volume.
• Charles's Law – the relationship between volume and temperature.
• Gay-Lussac's Law – the relationship between pressure and temperature.

Each of these laws helps to simplify the complex behaviors of gases by studying them under specific sets of conditions. For students, grasping these laws provides a baseline to solve various real-life problems that involve gases, like changes in air pressure or balloons' inflation and deflation.

Pressure-Volume Relationship

The relationship between pressure and volume is a core aspect of Boyle's Law, which states that for a fixed amount of gas at a constant temperature, the pressure of the gas is inversely proportional to its volume. This can be mathematically expressed as:\[ P \times V = ext{constant} \]What this means is when the pressure of a gas increases, its volume decreases if the temperature doesn't change, and vice versa. This relationship is especially important for understanding how gases behave in closed systems, such as tires or scuba diving tanks. It's crucial to keep in mind that this inverse relationship is why we can compress gases tightly into containers. Additionally, it explains why balloons expand when brought to higher altitudes, where atmospheric pressure is lower. This formula can be rearranged depending on the situation to solve for unknown values, making it a valuable tool for solving both theoretical and practical problems involving gases.

Inversely Proportional

The concept of inversely proportional is key in understanding Boyle's Law, which dictates that two quantities are inversely proportional if one increases while the other decreases. This may seem counterintuitive at first, but it's all about balancing both variables in the equation. When we say pressure and volume are inversely proportional, we're observing that as one gets larger, the other must get smaller to maintain balance. In the exercise above, this means that when pressure quadrupled from 1 atm to 4 atm, the volume dropped proportionally, shrinking from 100 cc to 25 cc. Visualizing this, you can think of squeezing a balloon; as you apply more pressure (by squeezing harder), the volume of the balloon diminishes. Understanding this concept is crucial not only for mathematical calculations but for grasping real-world phenomena, illustrating how energy is conserved in physical systems. This knowledge equips students to predict changes in systems where inversely proportional relationships are present, making educated decisions based on gas behavior insights.