Consider a balloon that has a volume V. It contains n moles of gas, it has an internal pressure of P, and its temperature is T. If the balloon is heated to a temperature of 15.5T while it is placed under a high pressure of 15.5P, how does the volume of the balloon change?
A. It doubles.
B. It stays the same.
C. It increases greatly.
D. It decreases slightly.
Answer (2)
The Ideal Gas Law states that P V = n RT .
The initial state is P V = n RT .
The final state is ( 15.5 P ) V f = n R ( 15.5 T ) .
Dividing the final state equation by the initial state equation and simplifying, we find that V f = V . Thus, the volume stays the same. I t s t a ys t h es am e .
Explanation
Problem Analysis We are given a balloon with initial volume V , number of moles n , pressure P , and temperature T . The balloon is then heated to a temperature of 15.5 T and placed under a pressure of 15.5 P . We want to determine how the volume of the balloon changes. We will use the Ideal Gas Law to relate the initial and final states of the gas in the balloon.
Ideal Gas Law The Ideal Gas Law is given by P V = n RT , where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature.
Initial State For the initial state, we have P V = n RT .
Final State For the final state, we have P f V f = n R T f , where P f = 15.5 P and T f = 15.5 T . Substituting these values, we get ( 15.5 P ) V f = n R ( 15.5 T ) .
Dividing the Equations Now, we divide the equation for the final state by the equation for the initial state: P V ( 15.5 P ) V f = n RT n R ( 15.5 T )
Simplifying Simplifying the equation, we have: 15.5 V V f = 15.5
Solving for Final Volume Dividing both sides by 15.5, we get: V V f = 1 Therefore, V f = V .
Conclusion The final volume V f is equal to the initial volume V . Therefore, the volume of the balloon stays the same.
Examples
The Ideal Gas Law is often used in real-world applications such as designing engines, understanding weather patterns, and even in scuba diving. For example, when designing an engine, engineers need to understand how the pressure, volume, and temperature of the gas inside the engine's cylinders change during the combustion process. By using the Ideal Gas Law, they can predict how the gas will behave under different conditions and optimize the engine's performance. Similarly, meteorologists use the Ideal Gas Law to understand how air masses behave in the atmosphere, which helps them predict weather patterns. In scuba diving, divers need to understand how the pressure of the air in their tanks changes as they descend into deeper water. The Ideal Gas Law helps them calculate how much air they need to breathe at different depths.
Using the Ideal Gas Law, we find that the volume of the balloon remains the same after being heated and placed under increased pressure. Therefore, the final volume is equal to the initial volume. The correct choice is B. It stays the same.
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