A bicycle tire has a volume of 1.0 L at $22^{\circ} C$. If the temperature of the tire is increased to $52^{\circ} C$, what is the resulting volume of the tire? Type in your answer using the correct number of significant figures.
Use the formula for Charles' Law: $\frac{V_1}{T_1}=\frac{V_2}{T_2}$
Answer (2)
Convert the temperatures from Celsius to Kelvin: T 1 = 22 + 273.15 = 295.15 K and T 2 = 52 + 273.15 = 325.15 K.
Apply Charles' Law to find the final volume: V 2 = V 1 ⋅ T 1 T 2 = 1.0 ⋅ 295.15 325.15 .
Calculate the final volume: V 2 = 1.1016432322547858 L.
Round the result to two significant figures: V 2 ≈ 1.1 L. The final volume of the tire is 1.1 L.
Explanation
Understanding the Problem We are given a problem relating the volume and temperature of a bicycle tire using Charles' Law. We have the initial volume V 1 = 1.0 L and initial temperature T 1 = 2 2 ∘ C . The temperature is increased to T 2 = 5 2 ∘ C , and we need to find the final volume V 2 . Charles' Law states that T 1 V 1 = T 2 V 2 .
Converting to Kelvin First, we need to convert the temperatures from Celsius to Kelvin because Charles' Law requires the temperature to be in Kelvin. The conversion formula is T ( K ) = T ( ∘ C ) + 273.15 . So, we have:
T 1 = 22 + 273.15 = 295.15 K T 2 = 52 + 273.15 = 325.15 K
Applying Charles' Law Now we can use Charles' Law to solve for V 2 :
T 1 V 1 = T 2 V 2 V 2 = V 1 ⋅ T 1 T 2 Substituting the given values, we get: V 2 = 1.0 ⋅ 295.15 325.15 V 2 = 1.0 ⋅ 1.1016432322547858 V 2 = 1.1016432322547858 L
Rounding to Significant Figures We need to round the answer to two significant figures because the initial volume is given to two significant figures (1.0 L). Therefore, V 2 ≈ 1.1 L
Final Answer The resulting volume of the tire is approximately 1.1 L.
Examples
Charles' Law is useful in understanding how gases behave under different temperatures, which has many real-world applications. For example, it helps explain why a hot air balloon rises. As the air inside the balloon is heated, its volume increases, making the balloon less dense than the surrounding air. This difference in density creates an upward buoyant force, causing the balloon to float. Similarly, understanding Charles' Law is crucial in designing engines and other systems that rely on the behavior of gases at varying temperatures.
Using Charles' Law, the volume of a bicycle tire increases from 1.0 L at 22°C to approximately 1.1 L at 52°C after converting temperatures to Kelvin. The calculations involve using the formula V 2 = V 1 ⋅ T 1 T 2 . The final volume is rounded to two significant figures due to the initial measurement.
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