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course PHY 232
2/12 1:24 pm
`q001. Suppose we have 10 moles of an ideal monatomic gas (a monatomic gas consists of single, unjoined atoms, and so consists of point masses only). If we are to increase the temperature of this gas by 240 degrees Kelvin, without expansion, how much energy do we need to put into the gas?****
`d E= `d KE_tr+ `d KE_rot+ `d W_P
Monatomic: KE_rot= 0
No Expansion: `d W_P = 0
3 deg of freedom:
`d E= 3(1/2)(nR `d T)= 3/2(10)(8.31)(240)= 29916 J
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To increase the temperature of 10 moles of ideal diatomic gas (a gas consisting of pairs of atoms), without expansion, how much energy do we need to put into the gas?
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Same as above but with 5 deg of freedom.
`d E= `d KE_rot + `d KE_tr = 2/2nR `d T + 3/2 nR `d T = 5/2 nR `d T
`d E= 5/2(10*8.31*240)= 49860 J
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If we put 5000 Joules of thermal energy into 10 moles of ideal monatomic gas, without expansion, by how much will the temperature change?
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`d E= 5000= 3/2nR `d T= 3/2*10*8.31* `d T
`d T= 40.11 K
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If we put 5000 Joules of thermal energy into 10 moles of ideal diatomic gas, without expansion, by how much will the temperature change?
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`d E= 5000= 5/2nR `d T= 5/2(10)(8.31) `d T
`d T= 24.07 K
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`q002. If we have a cubic meter of gas at atmospheric pressure and temperature 273 K, then that gas consists of about 45 moles. It doesn't matter whether it's monatomic or diatomic.
Suppose now that the gas is diatomic.
If we want to heat a cubic meter of the gas from 273 K to 373 K, without changing its volume, how much thermal energy is required?
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`d T= 100 K n= 45
`d E= 5/2nR `d T= 93488 J
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If we want to heat the gas, starting at 273 K and 1 atmosphere of pressure, at constant volume, until its pressure has increased by 50 000 N / m^2 (i.e., 50 000 Pa or 50 kPa), how much thermal energy will it take?
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P1/T1= P2/T2 ---> 100,000/273=150,000/T2 -->409.5 K
`d E= `d KE_tr= 5/2nR `d T---> `d E= 128 kJ
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How would this result change if the gas was monatomic?
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`d E= 3/2 nR `d T= 76.6 kJ
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`q003. What is the average translational KE per particle of a gas at temperature 300 K?
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KE/part = 3/2kT= 3/2(1.381^10^-23)*300= 6.215x10^-21
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What therefore would be the rms speed of particles of mass 1.66 * 10^-27 kg?
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v_rms= sqrt (3kT/m)= sqrt (2(KE/part)/m)= 2736 m/s
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What would be the rms speed of particles whose mass is 9 times as great as in the preceding?
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Sub 9m for m:
v_rms= sqrt (3kT/9m)= 1/3(v_rms_1m)= 912 m/s
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Diatomic oxygen molecules have mass about 32 times as great as 1.66 * 10^-27 kg. What is the rms speed of oxygen molecules in a gas at 300 K?
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Again sub 32m for m:
v_rms_O= sqrt 1/32(v_rms_1m) = 484 m/s
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What is the average rotational KE of oxygen molecules in a gas at 300 K?
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KE_rot = 2/3 KE_tr= 2/3(3/2kT)= 4.14x10^-21
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The effect of an elastic collisions between a particle and a receding wall is to reduce its speed. If the wall of a container is free to recede, then collisions with confined gas particles accelerate the wall, causing it to recede and reduce the speed of the particles. This causes the average KE of the particles to decrease, hence reducing the temperature of the gas.
If the wall is under pressure from the outside, it takes work to move it, and the work is done at the expense of the KE of the particles.
It is possible to add thermal energy to to the gas during the process, possibly mitigating the reduction of the KE of the particles (reducing the loss of temperature), possibly balancing the energy (hence keeping the temperature constant), perhaps even exceeding the energy required to move the wall (hence increasing the temperature).
The bottom line:
Any time a gas expands, it has to do work against pressure. Unless balanced by the input of thermal energy, the temperature of the gas will therefore decrease with expansion.
`q004. When we heat a gas at constant pressure, it expands, and some of the thermal energy added must do the work of expansion.
If a gas changes volume against constant pressure, then the thermal energy required of the gas is 2/2 n R `dT, which is conveniently equivalent to the thermal energy required for two additional degrees of freedom (there aren't two additional degrees of freedom, but the energy required is the same as if there were).
If the cubic meter of the diatomic gas in question 2 is heated from 273 K to 373 K at constant pressure, how much energy is required?
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`d E= `d KE_tr+ `d KE_rot+ `d W= 7/2 nRT= 131 kJ
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How much of that energy goes into changing the KE of the molecules, and how much into the work of expansion?
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`d W= nR `d T= 37.4 kJ
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How would these results change if the gas was monatomic?
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`d E= `d KE_tr+ `d W= 5/2 nRT= 93.5 kJ
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@& Everything looks good here.*@