Question

A gas thermometer contains 250 mL of gas at 0° C and 1.0 atm pressure. If the pressure remains at 1.0 atm, how many mL will the volume increase for every one celsius degree that the temperature rises?

I don't understand this part "for every one celsius degree"

Answer 1

Here's what I got.

Explanation:

The problem states that the pressure remains unchanged, so right from the start, you should know that you can use Charles' Law to calculate the change in volume associated with a `"1-"""^@"C"` increase in the temperature of the gas.

`color(blue)(ul(color(black)(V_1/T_1 = V_2/T_2)))`

Here

• `V_1` and `T_1` represent the volume and the absolute temperature of the gas at an initial state
• `V_2` and `T_2` represent the volume and the absolute temperature of the gas at a final state

Now, it's very important to remember that you must work with absolute temperatures here, i.e. with temperatures expressed in Kelvin.

In your case, the gas starts at

`0^@"C" = 0^@"C" + 273.15 = "273.15 K"`

When the temperature of the gas increases by `1^@"C"`, it also increases by `"1 K"` because you have

`0^@"C" + 1^@"C" = 1^@"C" ->` the temperatue increases by `1^@"C"`

which is

`1^@"C" = 1^@"C" + 273.15 = "274.15 K"`

Similarly, you will have

`"273.15 K" + "1 K" = "274.15 K" ->` the temperature increases by `"1 K"`

So, you must determine the change in volume that accompanies a `1^@"C" = "1 K"` increase in temperature. If you start at `0^@"C" = "273.15 K"`, you will end up at `"274.15 K"`.

Rearrange the equation to solve for `V_2`

`V_1/T_1 = V_2/T_2 implies V_2 = T_2/T_1 * V_1`

Plug in your values to find

`V_2 = (274.15 color(red)(cancel(color(black)("K"))))/(273.15color(red)(cancel(color(black)("K")))) * "250 mL" = "250.9 mL"`

So the volume of the gas increased by

`overbrace("250.9 mL")^(color(blue)("at 274.15 K" = 1^@"C")) - overbrace("250 mL")^(color(blue)("at 273.15 K" = 0^@"C")) = "0.9 mL"`

Notice what happens when you increase the temperature from `"274.15 K"` to `"275.15 K"`. You will once again have--remember to use the volume of the gas at `"274.15 mL"` as `V_1`

`V_2 = (275.15 color(red)(cancel(color(black)("K"))))/(274.15 color(red)(cancel(color(black)("K")))) * "250.9 mL" = "251.8 mL"`

The volume of the gas increased by

`overbrace("251.8 mL")^(color(blue)("at 275.15 K" = 2^@"C")) - overbrace("250.9 mL")^(color(blue)("at 274.15 K" = 1^@"C")) = "0.9 mL"`

You can thus say that with every `1^@"C" = "1 K"` increase in temperature, the volume of the gas increases by `"0.9 mL"`.

If you were to draw a graph of this relationship, you'd end up with a straight line that goes `"0.9 mL"` up as you move `"1 K"` to the right, i.e. the volume increases by `"0.9 mL"` with every `"1 K"` increase in temperature.

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