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A spherical balloon is filled with 4500 π cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 75 π cubic meters per minute, then the rate (in metres per minute) at which the radius of the balloon decreases 49 minutes after the leakage began is
- a)2/9
- b)9/2
- c)9/7
- d)7/9
Correct answer is option 'A'. Can you explain this answer?
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A spherical balloon is filled with 4500 π cubic meters of helium ga...
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A spherical balloon is filled with 4500 π cubic meters of helium ga...
Cubic centimeters of helium gas. If the radius of the balloon is 15 centimeters, what is the pressure of the gas inside the balloon?
We can use the ideal gas law, which states that:
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, and T is the temperature of the gas in Kelvin.
Since the balloon is spherical, we can calculate its volume using the formula for the volume of a sphere:
V = (4/3)πr^3
where r is the radius of the sphere.
Plugging in the given values, we get:
V = (4/3)π(15^3) = 14137.2 cubic centimeters
Next, we need to convert the volume of the gas from cubic centimeters to liters, since the ideal gas constant has units of liters-atmospheres per mole-kelvin. There are 1000 cubic centimeters in a liter, so:
V = 14.1372 liters
We also need to convert the temperature of the gas from Celsius to Kelvin, by adding 273.15:
T = 273.15 + 20 = 293.15 Kelvin
Now we can solve for the pressure of the gas, using the ideal gas law:
PV = nRT
n/V = P/RT (since n/V is the number of moles per liter)
P = (n/V)RT
We need to find n/V, the number of moles of gas per liter of volume. To do this, we can use the molar volume of an ideal gas at standard temperature and pressure, which is 22.4 liters per mole. Since the gas in the balloon is not at standard temperature and pressure, we need to adjust this value using the following formula:
molar volume = (V1/T1) * (T2/P2) * molar volume at STP
where V1 and T1 are the initial volume and temperature of the gas, and P2 is the pressure of the gas in the balloon. Solving for molar volume, we get:
molar volume = (14.1372/293.15) * (1/atm) * 22.4 = 0.864 liters/mole
Now we can find n/V:
n/V = (4500/1000) / 0.864 = 5.21 moles/liter
Finally, we can substitute into the ideal gas law:
P = (n/V)RT = (5.21)(0.08206)(293.15) = 121.4 kPa
Therefore, the pressure of the gas inside the balloon is approximately 121.4 kilopascals.
We can use the ideal gas law, which states that:
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, and T is the temperature of the gas in Kelvin.
Since the balloon is spherical, we can calculate its volume using the formula for the volume of a sphere:
V = (4/3)πr^3
where r is the radius of the sphere.
Plugging in the given values, we get:
V = (4/3)π(15^3) = 14137.2 cubic centimeters
Next, we need to convert the volume of the gas from cubic centimeters to liters, since the ideal gas constant has units of liters-atmospheres per mole-kelvin. There are 1000 cubic centimeters in a liter, so:
V = 14.1372 liters
We also need to convert the temperature of the gas from Celsius to Kelvin, by adding 273.15:
T = 273.15 + 20 = 293.15 Kelvin
Now we can solve for the pressure of the gas, using the ideal gas law:
PV = nRT
n/V = P/RT (since n/V is the number of moles per liter)
P = (n/V)RT
We need to find n/V, the number of moles of gas per liter of volume. To do this, we can use the molar volume of an ideal gas at standard temperature and pressure, which is 22.4 liters per mole. Since the gas in the balloon is not at standard temperature and pressure, we need to adjust this value using the following formula:
molar volume = (V1/T1) * (T2/P2) * molar volume at STP
where V1 and T1 are the initial volume and temperature of the gas, and P2 is the pressure of the gas in the balloon. Solving for molar volume, we get:
molar volume = (14.1372/293.15) * (1/atm) * 22.4 = 0.864 liters/mole
Now we can find n/V:
n/V = (4500/1000) / 0.864 = 5.21 moles/liter
Finally, we can substitute into the ideal gas law:
P = (n/V)RT = (5.21)(0.08206)(293.15) = 121.4 kPa
Therefore, the pressure of the gas inside the balloon is approximately 121.4 kilopascals.
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A spherical balloon is filled with 4500 π cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 75 π cubic meters per minute, then the rate (in metres per minute) at which the radius of the balloon decreases 49 minutes after the leakage began isa)2/9b)9/2c)9/7d)7/9Correct answer is option 'A'. Can you explain this answer?
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A spherical balloon is filled with 4500 π cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 75 π cubic meters per minute, then the rate (in metres per minute) at which the radius of the balloon decreases 49 minutes after the leakage began isa)2/9b)9/2c)9/7d)7/9Correct answer is option 'A'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A spherical balloon is filled with 4500 π cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 75 π cubic meters per minute, then the rate (in metres per minute) at which the radius of the balloon decreases 49 minutes after the leakage began isa)2/9b)9/2c)9/7d)7/9Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A spherical balloon is filled with 4500 π cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 75 π cubic meters per minute, then the rate (in metres per minute) at which the radius of the balloon decreases 49 minutes after the leakage began isa)2/9b)9/2c)9/7d)7/9Correct answer is option 'A'. Can you explain this answer?.
A spherical balloon is filled with 4500 π cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 75 π cubic meters per minute, then the rate (in metres per minute) at which the radius of the balloon decreases 49 minutes after the leakage began isa)2/9b)9/2c)9/7d)7/9Correct answer is option 'A'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A spherical balloon is filled with 4500 π cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 75 π cubic meters per minute, then the rate (in metres per minute) at which the radius of the balloon decreases 49 minutes after the leakage began isa)2/9b)9/2c)9/7d)7/9Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A spherical balloon is filled with 4500 π cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 75 π cubic meters per minute, then the rate (in metres per minute) at which the radius of the balloon decreases 49 minutes after the leakage began isa)2/9b)9/2c)9/7d)7/9Correct answer is option 'A'. Can you explain this answer?.
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