Sure, let's solve this problem step-by-step.
### Given Information:
- Initial pressure of the tire, [tex]\( P_1 = 3.00 \text{ atm} \)[/tex]
- Initial temperature of the tire, [tex]\( T_1 = 30.0^{\circ}C \)[/tex]
- Final temperature of the tire, [tex]\( T_2 = -5.00^{\circ}C \)[/tex]
- We need to find the final pressure, [tex]\( P_2 \)[/tex], assuming no volume change in the tire.
### Formula to Use:
The problem states that we should use the formula for the relationship between pressure and temperature when volume is constant:
[tex]\[
\frac{P_1}{T_1} = \frac{P_2}{T_2}
\][/tex]
### Step-by-Step Solution:
1. Convert Temperatures from Celsius to Kelvin:
Temperatures need to be in Kelvin for the formula to work correctly, as the Kelvin scale is the absolute temperature scale used in gas law equations.
[tex]\[
T_1(K) = T_1(^{\circ}C) + 273.15 = 30.0 + 273.15 = 303.15 \text{ K}
\][/tex]
[tex]\[
T_2(K) = T_2(^{\circ}C) + 273.15 = -5.00 + 273.15 = 268.15 \text{ K}
\][/tex]
2. Use the Formula to Find the Final Pressure:
Rearrange the equation [tex]\(\frac{P_1}{T_1} = \frac{P_2}{T_2}\)[/tex] to solve for [tex]\(P_2\)[/tex]:
[tex]\[
P_2 = P_1 \times \frac{T_2}{T_1}
\][/tex]
3. Substitute the Known Values:
[tex]\[
P_2 = 3.00 \text{ atm} \times \frac{268.15 \text{ K}}{303.15 \text{ K}}
\][/tex]
4. Calculate the Final Pressure:
[tex]\[
P_2 \approx 3.00 \times 0.884 = 2.65 \text{ atm}
\][/tex]
So, the final pressure in the tire, when the temperature decreases to [tex]\(-5.00^{\circ}C\)[/tex], is approximately [tex]\(2.65\)[/tex] atm. The intermediate values used, specifically the converted temperatures, are [tex]\(303.15\)[/tex] K and [tex]\(268.15\)[/tex] K.
