Air is added to a balloon containing 10 moles of air. It increases in size from .2 m3 to .3 m3
To solve this problem, we can use the ideal gas law equation:
PV = nRT
Where:
P = pressure (assume constant)
V = volume
n = moles of air
R = ideal gas constant
T = temperature (assume constant)
Since air is added to the balloon, the moles of air remain the same (10 moles). Therefore, we can rewrite the equation as:
P1V1 = P2V2
Where:
P1 = initial pressure
V1 = initial volume
P2 = final pressure (assume constant)
V2 = final volume
Rearranging the equation to solve for P2, we get:
P2 = (P1V1) / V2
Let's assume the initial pressure is constant at P1 = 1 atm. Substituting the given values:
P2 = (1 atm * 0.2 m^3) / 0.3 m^3
P2 = 0.67 atm
Therefore, the final pressure inside the balloon is approximately 0.67 atm.
To solve this problem, we can use the ideal gas law, which states:
PV = nRT
Where:
P = pressure (in pascals)
V = volume (in cubic meters)
n = number of moles
R = ideal gas constant (8.314 J/(mol·K))
T = temperature (in Kelvin)
We want to find the number of moles of air added to the balloon. Therefore, we need to calculate the initial number of moles and then subtract it from the final number of moles.
Step 1: Convert the initial and final volumes to liters:
Initial volume: 0.2 m^3 × 1000 L/m^3 = 200 L
Final volume: 0.3 m^3 × 1000 L/m^3 = 300 L
Step 2: Calculate the initial number of moles:
PV = nRT
n = PV / RT
Using the ideal gas law, we can rearrange the formula: n = PV / RT
Assuming standard temperature and pressure (STP), where T = 273.15 K and P = 101.325 kPa (or 101,325 Pa), we can calculate the initial number of moles:
n_initial = (P_initial × V_initial) / (R × T)
= (101325 Pa × 200 L) / (8.314 J/(mol·K) × 273.15 K)
Step 3: Calculate the final number of moles:
n_final = (P_final × V_final) / (R × T)
= (101325 Pa × 300 L) / (8.314 J/(mol·K) × 273.15 K)
Step 4: Calculate the number of moles added to the balloon:
n_added = n_final - n_initial
Now let's do the calculation:
n_initial = (101325 Pa × 200 L) / (8.314 J/(mol·K) × 273.15 K) = approximately 7.217 moles
n_final = (101325 Pa × 300 L) / (8.314 J/(mol·K) × 273.15 K) = approximately 10.826 moles
n_added = 10.826 moles - 7.217 moles = approximately 3.609 moles
Therefore, approximately 3.609 moles of air were added to the balloon.
To find the total number of moles of air added to the balloon, we need to use the ideal gas law equation.
The ideal gas law equation is: PV = nRT
Where:
P is the pressure of the gas
V is the volume of the gas
n is the number of moles of the gas
R is the ideal gas constant
T is the temperature of the gas (in Kelvin)
In this case, the pressure, temperature, and ideal gas constant are constant. Therefore, we can rewrite the equation as:
V1/n1 = V2/n2
Where:
V1 is the initial volume of the gas (0.2 m^3)
V2 is the final volume of the gas (0.3 m^3)
n1 is the initial number of moles of the gas (10 moles)
n2 is the final number of moles of the gas (unknown)
Rearranging the equation, we can solve for n2:
n2 = (V2 * n1) / V1
Substituting the given values:
n2 = (0.3 m^3 * 10 moles) / 0.2 m^3
n2 = 1.5 * 10 moles
Therefore, the total number of moles of air added to the balloon is 15 moles.