Consider a system consisting of a cylinder with a movable piston containing 10^6 gas molecules at 298K at a volume of 1L. Consider the following descriptions of this system:
1. Initial system
2. Starting from the initial system, the volume of the container is changed to 2L and the temperature to 395K.
3. Starting from the initial system, a combination reaction occurs at constant volume and temperature.
4. Starting from the initial system, the gas reacts completely to produce 10^7 gas molecules at 395K in a volume of 2L.
5. Starting from the initial system, the temperature is changed to 395K at constant volume.
Arrange the descriptions in order of increasing number of microstates in the resulting system.
D B E A C
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To determine the order of increasing number of microstates in the resulting systems, we need to understand the concept of microstates and how they relate to the given descriptions.
Microstates refer to the different possible arrangements of the molecules within a system, considering their positions and speeds. In a system with gas molecules, microstates can be thought of as different combinations of the molecules' positions and velocities.
Let's analyze each description and determine the order of increasing number of microstates:
1. Initial system:
In this case, we have 10^6 gas molecules in a volume of 1L at 298K. The number of microstates can be calculated using the formula:
Ω = (V^N)/N!
Where Ω is the number of microstates, V is the volume, and N is the number of molecules.
So, for the initial system, Ω = (1L^10^6)/(10^6!) ≈ 10^10^6
2. Starting from the initial system, the volume of the container is changed to 2L, and the temperature to 395K:
In this case, both the volume and temperature of the system change. However, since the number of gas molecules remains the same, the number of microstates will also remain the same. Therefore, the number of microstates would still be approximately 10^10^6.
3. Starting from the initial system, a combination reaction occurs at constant volume and temperature:
In a combination reaction, the reactants combine to form new products. Since the volume and temperature remain constant, the only change is the rearrangement of the gas molecules due to the reaction. The number of molecules does not change, so the number of microstates will be the same as the initial system, i.e., approximately 10^10^6.
4. Starting from the initial system, the gas reacts completely to produce 10^7 gas molecules at 395K in a volume of 2L:
In this case, the reaction causes a change in both the number of molecules and the volume of the system. The number of microstates can be calculated using the same formula:
Ω = (V^N)/N!
For the resulting system, Ω = (2L^10^7)/(10^7!) ≈ 10^10^7
5. Starting from the initial system, the temperature is changed to 395K at constant volume:
In this case, only the temperature of the system changes, while the volume and the number of gas molecules remain constant. Changing the temperature only affects the kinetic energy and velocity distribution of the gas molecules, but it does not change the number of microstates. Therefore, the number of microstates would still be approximately 10^10^6.
Now, let's arrange the descriptions in order of increasing number of microstates:
1. Initial system (approximately 10^10^6 microstates)
2. Starting from the initial system, the volume of the container is changed to 2L, and the temperature to 395K (approximately 10^10^6 microstates)
3. Starting from the initial system, a combination reaction occurs at constant volume and temperature (approximately 10^10^6 microstates)
4. Starting from the initial system, the temperature is changed to 395K at constant volume (approximately 10^10^6 microstates)
5. Starting from the initial system, the gas reacts completely to produce 10^7 gas molecules at 395K in a volume of 2L (approximately 10^10^7 microstates)
Therefore, the correct order from least to greatest number of microstates is:
1. Initial system
2. Starting from the initial system, the volume of the container is changed to 2L, and the temperature to 395K.
3. Starting from the initial system, a combination reaction occurs at constant volume and temperature.
4. Starting from the initial system, the temperature is changed to 395K at constant volume.
5. Starting from the initial system, the gas reacts completely to produce 10^7 gas molecules at 395K in a volume of 2L.