An automobile tire at $30.0^{\circ} C$ has a pressure of 3.00 atm. The temperature decreases to $-5.00^{\circ} C$. Assume that there is no volume change in the tire.
Formula to use: $\frac{P_1}{T_1}=\frac{P_2}{T_2}$
How does the tire pressure change in response to the temperature change?
Answer (2)
The tire pressure decreases from 3.00 atm to approximately 2.65 atm as the temperature drops from 30.0°C to -5.0°C. This relationship is explained by Gay-Lussac's Law, which states that pressure and temperature are directly related when volume is constant. Thus, a decrease in temperature leads to a decrease in pressure in the tire.
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Convert the temperatures from Celsius to Kelvin: T 1 = 30.0 + 273.15 = 303.15 K and T 2 = − 5.00 + 273.15 = 268.15 K .
Use the formula T 1 P 1 = T 2 P 2 to relate initial and final pressures and temperatures.
Solve for the final pressure: P 2 = P 1 × T 1 T 2 = 3.00 a t m × 303.15 K 268.15 K .
Calculate the final pressure: P 2 ≈ 2.65 a t m .
Explanation
Understanding the Problem We are given the initial temperature and pressure of an automobile tire, and we are asked to find the final pressure after the temperature decreases, assuming the volume remains constant. We will use the formula T 1 P 1 = T 2 P 2 to solve for the final pressure P 2 .
Converting Temperatures to Kelvin First, we need to convert the temperatures from Celsius to Kelvin. The conversion formula is T ( K ) = T ( ∘ C ) + 273.15 . Thus,
T 1 = 30. 0 ∘ C + 273.15 = 303.15 K
T 2 = − 5.0 0 ∘ C + 273.15 = 268.15 K
Isolating the Final Pressure Now we use the formula T 1 P 1 = T 2 P 2 to solve for P 2 . We have P 1 = 3.00 a t m , T 1 = 303.15 K , and T 2 = 268.15 K . Rearranging the formula to isolate P 2 , we get:
P 2 = P 1 × T 1 T 2
Calculating the Final Pressure Substituting the values, we have:
P 2 = 3.00 a t m × 303.15 K 268.15 K = 2.6536 a t m
Rounding to three significant figures, we get P 2 = 2.65 a t m .
Final Answer The final pressure of the tire is approximately 2.65 atm. The tire pressure decreases in response to the temperature decrease.
Examples
Understanding how pressure changes with temperature is crucial in many real-world applications. For example, knowing how tire pressure changes with temperature helps ensure safe driving conditions. If the temperature drops significantly, the tire pressure decreases, which can affect the car's handling and fuel efficiency. Similarly, this principle is used in designing pressure vessels and understanding weather patterns.
