Suppose you double the temperature of a gas at constant volume. Do the following change? If so, by what factor?
a. The average translational kinetic energy of a mole cule.
b. The rms speed of a molecule.
c. The mean free path.
(a) increase to double the initial value,
(b) increase to times the initial value,
(c) will not change if final pressure is doubled, twice if the final pressure is the same, otherwise cannot say.
(a) TThe equation is as follows for the average translational energy and temperature relationship:
Since everything apart from the temperature is a constant.
(b) The rms speed is given as
where we have collected all the constants under . This result means that the rms speed is proportional to the square root of the temperature. That is to say,
if the temperature is doubled, the rms speed will increase by the square root of two.
The formula of mean free path is
At first glance, the mean free path appears to be temperature agnostic. The numerical density, on the other hand, is temperature-dependent..
From the optmial gas law we have
The numerical density will fall by a factor of two if the temperature is doubled while the volume and pressure remain constant. However, until we know the end pressure, we can't say anything about the numerical density if the heating isn't also isobaric.
9. Suppose you place an ice cube in a beaker of room-temperature water, then seal them in a rigid, well-insulated container. No energy can enter or leave the container.
a. If you open the container an hour later, will you find a beaker of water slightly cooler than room temperature, or a large ice cube and some steam?
b. Finding a large ice cube and some steam would not violate the first law of thermodynamics. and because the container is sealed, and because the increase in thermal energy of the water molecules that became steam is offset by the decrease in thermal energy of the water molecules that turned to ice. Energy would be conserved, yet we never see an outcome like this. Why not?
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