For an ideal gas where the pressure
and volume are held constant, which of the following is the correct proportionality
relation between the temperature, 𝑇, of the gas and the number of moles, 𝑛? (A) 𝑇 is proportional to 𝑛. (B) 𝑇 is proportional to one over
𝑛. (C) 𝑇 is proportional to 𝑛
squared. (D) 𝑇 is proportional to one over
𝑛 squared. (E) 𝑇 is proportional to the
square root of 𝑛.
We’re told here that the gas we’re
considering is an ideal gas. Therefore, it can be described by
the ideal gas law. This law says that the pressure of
an ideal gas multiplied by its volume is equal to the number of moles of that gas
times a constant multiplied by the gas’s temperature. Here, we want to figure out the
relationship between that temperature 𝑇 and the number of moles of the gas 𝑛. To do this, let’s rearrange the
ideal gas law so that 𝑇 is the subject. If we divide both sides by 𝑛 times
𝑅, then both the number of moles 𝑛 and the gas constant 𝑅 cancel on the
right. And we find that the temperature 𝑇
is equal to the pressure of the gas times its volume divided by 𝑛, the number of
moles, times 𝑅, the molar gas constant.
The molar gas constant 𝑅 is always
a constant. In this particular case, we’re told
that the pressure and volume of our gas are held constant. Therefore, in this scenario, the
entire quantity 𝑃 times 𝑉 divided by 𝑅 is a constant. We could represent this quantity
using a symbol. Let’s use 𝑘. Our equation then becomes 𝑇 is
equal to the constant 𝑘 divided by the number of moles of the gas 𝑛. A mathematically equivalent way of
writing this is to say that 𝑇 is proportional to one over 𝑛. In other words, 𝑇 is inversely
proportional to 𝑛. Among our answer options, this
agrees with the option listed as choice (B).
For an ideal gas, where the
pressure and volume are held constant, the temperature 𝑇 of the gas is inversely
proportional to the number of moles of the gas 𝑛.