Question

Question #265c0

Answer 1

Here's how you can do that.

Explanation:

Your starting point here will be the ideal gas law equation, which looks like this

`color(blue)(ul(color(black)(PV = nRT)))`

Here

• `P` is the pressure of the gas
• `V` is the volume it occupies
• `n` is the number of moles of gas present in the sample
• `R` is the universal gas constant, equal to `0.0821("atm L")/("mol K")`
• `T` is the absolute temperature of the gas

Now, let's say that the given mass of gas is `m`. The number of moles of gas present in this given mass `m` depends on the molar mass of the gas, let's say `M_M`.

`n = m /M_M`

Plug this into the ideal gas law equation to get

`PV = m/M_M * RT`

Next, divide both sides of the equation by `T` to get

`(PV)/T = m/M_M * R`

The molar mass of the gas, which tells you the mass of exactly `1` mole of the gas, is constant.

`(PV)/T = overbrace(R/M_M)^(color(blue)("constant")) * m`

`(PV)/T = color(blue)("constant") * m`

This means that for a given mass `m`, i.e. if you use a sample of gas of constant mass, you can say that

`(PV)/T = color(blue)("constant")`

This is the combined gas law equation and it tells you that for a given mass of gas `m`, you have

`color(blue)(ul(color(black)((P_1V_1)/T_1 = (P_2V_2)/T_2)))`

Here

• `P_1`, `V_1`, `T_1` are the pressure, volume, and absolute temperature of the gas at an initial state
• `P_2`, `V_2`, `T_2` are the pressure, volume, and absolute temperature of the gas at a final state