Answer:
To find the initial quantity of helium present in the balloon before it expanded, you can use the ideal gas law equation:
PV = nRT
Where:
P is the pressure (which we'll assume remains constant).
V is the volume of the gas.
n is the number of moles of gas.
R is the ideal gas constant.
T is the absolute temperature.
In your case, the initial volume (V1) is 230 mL, and the final volume (V2) is 860 mL. The number of moles of helium (n) is given as 3.8 x 10 mol.
First, you need to convert the volumes to liters because the ideal gas constant R is typically used with SI units:
V1 = 230 mL = 0.230 L
V2 = 860 mL = 0.860 L
Now, you can use the ideal gas law to find the initial quantity of helium:
P * V1 = n * R * T1
P * V2 = n * R * T2
Since the pressure (P), the number of moles (n), and the ideal gas constant (R) remain constant, you can set the two equations equal to each other:
V1 / T1 = V2 / T2
Now, solve for T1:
T1 = (V1 * T2) / V2
T1 = (0.230 L * T2) / 0.860 L
T1 ≈ 0.061 * T2
Now, you have the initial temperature (T1) in terms of the final temperature (T2).
Next, you can use the relationship between temperature and the number of moles using the ideal gas law:
PV = nRT
T1 = (P * V1) / (n * R)
T2 = (P * V2) / (n * R)
Now, let's find T2:
T2 = (P * 0.860 L) / (3.8 x 10 mol * R)
T2 ≈ (P * 0.860 L) / (3.8 x 10 mol * 8.314 J/(mol·K)) (using the ideal gas constant R ≈ 8.314 J/(mol·K))
T2 ≈ (P * 0.860 L) / (31.6092 J/K)
Now, you have T2 in terms of pressure (P).
Now, substitute this expression for T2 back into the equation for T1:
T1 ≈ 0.061 * [(P * 0.860 L) / (31.6092 J/K)]
Now, we can find the initial quantity of helium (n1) using the initial temperature T1:
n1 = (P * V1) / (R * T1)
n1 = (P * 0.230 L) / (8.314 J/(mol·K) * 0.061 * [(P * 0.860 L) / (31.6092 J/K)])
Now, calculate n1:
n1 ≈ (0.230 * P) / (0.506244 * P)
n1 ≈ 0.4544 * P
Now, we know that the expanded balloon contains 3.8 x 10 mol of helium, so:
n1 = 0.4544 * P = 3.8 x 10 mol
Now, solve for P:
P = (3.8 x 10 mol) / 0.4544
P ≈ 8.36 x 10 mol
Now that we have found the pressure (P), we can find the initial quantity of helium (n1):
n1 ≈ 0.4544 * (8.36 x 10 mol)
n1 ≈ 3.8 x 10 mol
So, the initial quantity of helium present in the balloon was approximately 3.8 x 10 moles.
Step-by-step explanation: