Final answer:
The gauge pressure in the car tire at 25.0°C with 3.41 mol of gas in a 28.0 L volume is 2.76 atm. After adding 1.50 L of gas at atmospheric pressure, the gauge pressure increases slightly to 2.87 atm.
Step-by-step explanation:
Calculating Gauge Pressure in a Car Tire
To calculate the gauge pressure in the car tire, we will use the Ideal Gas Law, which states that PV=nRT, where P is the pressure, V is the volume, n is the number of moles of gas, R is the universal gas constant, and T is the temperature in Kelvin.
(a) First, we convert the temperature to Kelvin: T = 25.0 + 273.15 = 298.15 K. Then we can calculate the gauge pressure using:
P = (nRT)/V
P = (3.41 mol × 0.0821 L·atm/K·mol × 298.15 K) / 28.0 L
P = (3.41 × 0.0821 × 298.15) / 28.0 = 2.76 atm
(b) If we add 1.50 L of gas at atmospheric pressure (1 atm) without changing the volume, we have an additional number of moles n' calculated by n' = PV/RT, so:
n' = (1 atm × 1.50 L) / (0.0821 L·atm/K·mol × 298.15 K)
n' = 1.50 / (0.0821 × 298.15)
n' = 0.0618 mol
We then add this to the original number of moles and recalculate the pressure:
P' = ((3.41 + 0.0618) mol × 0.0821 L·atm/K·mol × 298.15 K) / 28.0 L
P' = (3.4718 × 0.0821 × 298.15) / 28.0 = 2.87 atm