Question

Many drivers report low tire pressure at the start of winter when the temperature drops. If the pressure in a tire is 2. 38 atm when the outside temperature is 294 k , what will the pressure in the tire be if the outside temperature drops to 280. K ? (assume that the volume and atmospheric pressure remain constant. ).

Answer 1

Approximately
`2.27\; {\rm atm}`.

Step-by-step explanation:

Assuming that the air in the tire is an ideal gas. The ideal gas law will apply:

`P\, V = n\, R\, T`,

Where:

• 
  `P` is the pressure in the tire,
• 
  `V` is the volume of the air in the tire,
• 
  `n` is the quantity of air particles in the tire,
• 
  `R` is the ideal gas constant, and
• 
  `T` is the temperature of the air in the tire.

Rearrange this equation to keep pressure
`P` and temperature
`T` on the same side of the equality:

`\displaystyle (P)/(T) = (n\, R)/(V)`.

The value of
`R` is a constant. Under the assumptions, the value of
`n` and
`V` would also be constant. In other words, the value of
`(n\, R / V)` would stay constant.

Let
`P_(1) = 2.38\; {\rm atm}` and
`T_(1) = 294\; {\rm K}` denote the initial pressure and temperature; let
`P_(2)` and
`T_(2) = 280\; {\rm K}` denote these two quantities after the temperature change. Under the assumptions:

`\displaystyle (P_(1))/(T_(1)) = (n\, R)/(V) = (P_(2))/(T_(2))`.

Rearrange this equation to obtain an expression for the new pressure in the tire,
`P_(2)`:

`\begin{aligned}P_(2) &= (P_(1)\, T_(2))/(T_(1)) \\ &= (2.38\; {\rm atm})\, \left(\frac{280\; {\rm K}}{294\; {\rm K}}\right) \\ &\approx 2.27\; {\rm atm}\end{aligned}`.

In other words, the pressure of the air inside the tire would be approximately
`2.27\; {\rm atm}` after the temperature change.