What is the temperature of 0.750 mol of a gas stored in a [tex]$6,850 mL$[/tex] cylinder at 2.21 atm? Use [tex]$P V=n R T$[/tex] and [tex]$R=0.0821 \frac{L \bullet \text { atrm }}{ mol \bullet K }$[/tex].
A. 2.95 K
B. 5.24 K
C. 138 K
D. 246 K
Answer (2)
Convert the volume from mL to L: V = 6850 mL = 6.85 L .
Apply the Ideal Gas Law formula: T = n R P V .
Substitute the given values: T = ( 0.750 mol ) ( 0.0821 mol ⋅ K L ⋅ atm ) ( 2.21 atm ) ( 6.85 L ) .
Calculate the temperature: T ≈ 246 K .
The temperature of the gas is 246 K .
Explanation
Problem Setup and Given Information We are given the following information:
Number of moles of gas, n = 0.750 mol
Volume of the gas cylinder, V = 6 , 850 mL
Pressure of the gas, P = 2.21 atm
Ideal gas constant, R = 0.0821 mol ⋅ K L ⋅ atm
We need to find the temperature T of the gas in Kelvin using the Ideal Gas Law: P V = n RT .
Volume Conversion First, we need to convert the volume from milliliters (mL) to liters (L) since the ideal gas constant R is given in terms of liters. To convert mL to L, we divide by 1000:
V ( L ) = 1000 V ( mL ) = 1000 6850 = 6.85 L
So, the volume of the gas is 6.85 L .
Applying the Ideal Gas Law Now, we can use the Ideal Gas Law, P V = n RT , to solve for the temperature T . We rearrange the equation to isolate T :
T = n R P V
Next, we substitute the given values into the equation:
T = ( 0.750 mol ) ( 0.0821 mol ⋅ K L ⋅ atm ) ( 2.21 atm ) ( 6.85 L )
Calculating the Temperature Now, we perform the calculation:
T = 0.750 × 0.0821 2.21 × 6.85 = 0.061575 15.1385 ≈ 245.85 K
Therefore, the temperature of the gas is approximately 245.85 K .
Final Answer and Conclusion The temperature of the gas is approximately 245.85 K . Among the given options, the closest value is 246 K .
Examples
The Ideal Gas Law is incredibly useful in various real-world scenarios. For instance, it helps predict the behavior of gases in weather forecasting, where understanding temperature, pressure, and volume relationships is crucial. In the automotive industry, it aids in designing efficient engines by optimizing fuel combustion. Moreover, in scuba diving, the Ideal Gas Law is essential for calculating how gases behave at different depths, ensuring divers' safety by predicting changes in tank volume and pressure.
Using the Ideal Gas Law, we calculated the temperature of the gas to be approximately 246 K. This was determined by converting the volume from mL to L, rearranging the Ideal Gas Law, and substituting the given values. The closest answer from the options is 246 K.
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