I need help on these two questions:

If a certain gas occupies a volume of 16L when the applied pressure is 8.0atm , find the pressure when the gas occupies a volume of 4.0L .

A certain gas is present in a 12.0L cylinder at 4.0atm pressure. If the pressure is increased to 8.0atm the volume of the gas decreases to 6.0L . Find the two constants k1, the initial value of k, and kf, the final value of k, to verify whether the gas obeys Boyle’s law.

#1.

P1V1 = P2V2

#2.
P1V1 = k1
P2V2 = k2

Sure, I can help you with these questions step-by-step.

Question 1: If a certain gas occupies a volume of 16L when the applied pressure is 8.0atm, find the pressure when the gas occupies a volume of 4.0L.

To solve this question, we can use Boyle's Law, which states that the volume of a gas is inversely proportional to its pressure when temperature is constant. Boyle's Law can be expressed as:

P1 * V1 = P2 * V2,

where P1 and P2 are the initial and final pressures, and V1 and V2 are the initial and final volumes respectively.

Given:
P1 = 8.0 atm,
V1 = 16 L,
V2 = 4.0 L.

Using the Boyle's Law equation, we can solve for P2:

P1 * V1 = P2 * V2
8.0 atm * 16 L = P2 * 4.0 L

Simplifying the equation:
(8.0 atm * 16 L) / 4.0 L = P2

P2 = 32.0 atm

Therefore, the pressure when the gas occupies a volume of 4.0L is 32.0 atm.

Question 2: A certain gas is present in a 12.0L cylinder at 4.0 atm pressure. If the pressure is increased to 8.0 atm, the volume of the gas decreases to 6.0L. Find the two constants k1, the initial value of k, and kf, the final value of k, to verify whether the gas obeys Boyle's law.

Boyle's Law states that the product of the initial pressure and initial volume of a gas is equal to the product of the final pressure and final volume, as long as the temperature remains constant.

Mathematically, this can be expressed as:

P1 * V1 = P2 * V2,

where P1 and P2 are the initial and final pressures, and V1 and V2 are the initial and final volumes respectively.

Given:
P1 = 4.0 atm,
P2 = 8.0 atm,
V1 = 12.0 L,
V2 = 6.0 L.

Using the Boyle's Law equation, we can solve for the two constants k1 and kf:

P1 * V1 = P2 * V2
4.0 atm * 12.0 L = 8.0 atm * 6.0 L

Simplifying the equation:
k1 = (4.0 atm * 12.0 L) / (8.0 atm * 6.0 L)
k1 = 48.0 L/atm

kf = k1 = 48.0 L/atm

Since k1 = kf, the gas in this scenario obeys Boyle's Law.

I hope this helps! Let me know if you have any other questions.

To solve the first question, you can use Boyle's law, which states that the pressure and volume of a gas are inversely proportional when the temperature is constant.

Boyle's law equation: P1 * V1 = P2 * V2

Where:
P1 = Initial pressure
V1 = Initial volume
P2 = Final pressure
V2 = Final volume

Given:
P1 = 8.0atm
V1 = 16L
V2 = 4.0L

Substituting the given values into the Boyle's law equation:
8.0atm * 16L = P2 * 4.0L

Now we can solve for P2:
(8.0atm * 16L) / 4.0L = P2

Simplifying the equation:
32 = P2

Therefore, the pressure when the gas occupies a volume of 4.0L is 32 atm.

For the second question, we need to verify if the gas obeys Boyle's law. According to Boyle's law, the product of pressure and volume should be constant when the temperature is constant.

Boyle's Law equation: P1 * V1 = P2 * V2

Given:
P1 = 4.0atm
V1 = 12.0L
P2 = 8.0atm
V2 = 6.0L

Substitute these values into the Boyle's law equation:
(4.0atm * 12.0L) = (8.0atm * 6.0L)

Simplifying the equation:
48.0 = 48.0

Since the two sides of the equation are equal, it means that the gas obeys Boyle's law.

To find the constants k1 and kf, we can rewrite the equation using the general form of Boyle's law:

P1 * V1 = P2 * V2 = k

Therefore, k1 = P1 * V1 and kf = P2 * V2.

Using the given values:
k1 = 4.0atm * 12.0L = 48.0
kf = 8.0atm * 6.0L = 48.0

Both k1 and kf are equal to 48.0, which verifies that the gas follows Boyle's law.

48,48