Video Transcript
For an ideal gas where the volume
and temperature are held constant, which of the following is the correct
proportionality relation between the pressure, 𝑃, of the gas and the number of
moles, 𝑛? (A) 𝑃 is proportional to one
divided by 𝑛. (B) 𝑃 is proportional to one
divided by 𝑛 squared. (C) 𝑃 is proportional to 𝑛
squared. (D) 𝑃 is proportional to 𝑛. (E) 𝑃 is proportional to the
square root of 𝑛.
We’re told that we’re working in
this scenario with an ideal gas. This gas can be described then by
the ideal gas law. This law says that for an ideal
gas, the gas’s pressure times its volume is equal to the number of moles of the gas
multiplied by a constant times the gas’s temperature. We’re interested in the
relationship between the pressure, 𝑃, and the number of moles of the gas, 𝑛. To make that relationship clearer,
let’s divide both sides of this equation by the volume of the gas, 𝑉, which cancels
that factor out on the left. This gives us an expression where
the pressure, 𝑃, is the subject.
On the right side of this equation,
the symbol 𝑅 represents a constant. And we’re told in our problem
statement that the gas volume and temperature are held constant. Therefore, 𝑅 times 𝑇 divided by
𝑉 is all equal to a constant value. If we wanted to, we could give a
name to this value. Say we call it capital 𝐶. In that case, our ideal gas law
equation can be written as 𝑃, the pressure, is equal to a constant, 𝐶, multiplied
by the number of moles of gas, 𝑛.
A mathematically equivalent way to
write this equation is to say that 𝑃 is proportional to 𝑛. This is true because once we have
𝑛, we just multiply it by a constant value to get the pressure, 𝑃. 𝑃 is proportional to 𝑛. And we see that this agrees with
one of our answer options. Answer choice (D) correctly says
that the proportionality relation between the pressure 𝑃 of the gas and the number
of moles 𝑛 of the gas is that 𝑃 is directly proportional to 𝑛.
