Calculate: Use the ideal gas law (PV = nRT) to solve the following. Show work for each problem. Then use the Gizmo to check your answer.
A. What is the volume of 0.5 moles of gas at STP?
B. How much pressure would 0.8 moles of a gas at 370 K exert if it occupied 17.3 L of space? V=nRT (0.8)(0.082)(370 K) P= nRT (0.8)(0.082)(370 K)/17.3 L p = 1.4 atm
C. How much H2 gas is necessary to exert a pressure of 1.4 atm at 430 K if occupying a volume of 15.1 L? n= PV/RT
The correct answer and explanation is :
Let’s go step by step to solve each problem using the ideal gas law ( PV = nRT ), where:
- ( P ) is pressure (in atm or Pa),
- ( V ) is volume (in liters or m³),
- ( n ) is the number of moles of gas,
- ( R ) is the ideal gas constant (( 0.0821 \, \text{L·atm/mol·K} )),
- ( T ) is temperature (in Kelvin, K).
A. What is the volume of 0.5 moles of gas at STP?
At STP (Standard Temperature and Pressure), the temperature is 273.15 K (0°C), and the pressure is 1 atm. We can rearrange the ideal gas law to solve for the volume ( V ):
[
V = \frac{nRT}{P}
]
Given:
- ( n = 0.5 \, \text{mol} ),
- ( R = 0.0821 \, \text{L·atm/mol·K} ),
- ( T = 273.15 \, \text{K} ),
- ( P = 1 \, \text{atm} ).
Now, substitute these values into the equation:
[
V = \frac{(0.5)(0.0821)(273.15)}{1}
]
[
V = \frac{11.217}{1} = 11.217 \, \text{L}
]
Thus, the volume of 0.5 moles of gas at STP is 11.22 L (rounded to two decimal places).
B. How much pressure would 0.8 moles of a gas at 370 K exert if it occupied 17.3 L of space?
Rearranging the ideal gas law to solve for pressure ( P ):
[
P = \frac{nRT}{V}
]
Given:
- ( n = 0.8 \, \text{mol} ),
- ( R = 0.0821 \, \text{L·atm/mol·K} ),
- ( T = 370 \, \text{K} ),
- ( V = 17.3 \, \text{L} ).
Now, substitute these values into the equation:
[
P = \frac{(0.8)(0.0821)(370)}{17.3}
]
[
P = \frac{24.3384}{17.3} = 1.41 \, \text{atm}
]
Thus, the pressure exerted by the gas is 1.4 atm (rounded to one decimal place).
C. How much H2 gas is necessary to exert a pressure of 1.4 atm at 430 K if occupying a volume of 15.1 L?
We need to solve for the number of moles ( n ) using the rearranged ideal gas law:
[
n = \frac{PV}{RT}
]
Given:
- ( P = 1.4 \, \text{atm} ),
- ( V = 15.1 \, \text{L} ),
- ( R = 0.0821 \, \text{L·atm/mol·K} ),
- ( T = 430 \, \text{K} ).
Now, substitute these values into the equation:
[
n = \frac{(1.4)(15.1)}{(0.0821)(430)}
]
[
n = \frac{21.14}{35.243} = 0.599 \, \text{mol}
]
Thus, the amount of H₂ gas necessary to exert a pressure of 1.4 atm at 430 K in 15.1 L is 0.599 moles.
Explanation
The ideal gas law is a powerful tool in chemistry that describes the behavior of gases under various conditions. It combines the relationship between pressure, volume, temperature, and the number of moles of a gas into a single equation. The three problems above demonstrate how to use this law to solve for different variables such as volume, pressure, and the number of moles.
In the first problem, we used the standard conditions of temperature and pressure (STP) to determine the volume occupied by 0.5 moles of gas. At STP, we expect gases to follow predictable behavior, which is why the calculation gave a reasonable result.
The second problem showed how pressure changes when moles and temperature are kept constant but the volume changes. Since the volume was given and temperature was elevated, the pressure came out to be a moderate value.
Finally, in the third problem, we applied the ideal gas law to calculate the number of moles required to achieve a specific pressure under certain conditions of temperature and volume.
In all cases, the ideal gas law is used to simplify and quantify the behavior of gases, providing a straightforward approach to understanding gas properties.