A fixed quantity of gas at 27°C exhibits a pressure of 729 torr and occupies a volume of 5.70 L.

(a) Use Boyle's law to calculate the volume the gas will occupy if the pressure is increased to 1.75 atm while the temperature is held constant.

(b) Use Charles's law to calculate the volume the gas will occupy if the temperature is increased to 154°C while the pressure is held constant.

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To solve these problems, we can use Boyle's law and Charles's law, which are two gas laws that relate temperature, pressure, and volume for a fixed amount of gas.

(a) Boyle's law states that the pressure of a gas is inversely proportional to its volume when the temperature is held constant. Mathematically, it can be written as:

P1 * V1 = P2 * V2

where P1 and V1 are the initial pressure and volume, and P2 and V2 are the final pressure and volume respectively.

Given:
Initial pressure, P1 = 729 torr
Initial volume, V1 = 5.70 L
Final pressure, P2 = 1.75 atm (we need to convert it to torr)
Final volume, V2 = ? (to be determined)

To convert the final pressure from atm to torr, we can use the conversion factor: 1 atm = 760 torr.

1.75 atm * 760 torr/atm = 1330 torr

Now we have the values:
P1 = 729 torr
V1 = 5.70 L
P2 = 1330 torr
V2 = ?

Using Boyle's law equation, we can write:

P1 * V1 = P2 * V2

Substituting the known values:

(729 torr) * (5.70 L) = (1330 torr) * V2

Solving for V2:

V2 = (729 torr * 5.70 L) / (1330 torr)
V2 ≈ 3.14 L

Therefore, if the pressure is increased to 1.75 atm while the temperature is held constant, the gas will occupy a volume of approximately 3.14 L.

(b) Charles's law states that the volume of a gas is directly proportional to its temperature when the pressure is held constant. Mathematically, it can be written as:

V1 / T1 = V2 / T2

where V1 and T1 are the initial volume and temperature, and V2 and T2 are the final volume and temperature respectively.

Given:
Initial volume, V1 = 5.70 L
Initial temperature, T1 = 27°C (we need to convert it to Kelvin)
Final volume, V2 = ? (to be determined)
Final temperature, T2 = 154°C (we need to convert it to Kelvin)

To convert temperature from Celsius to Kelvin, we can use the formula: T(K) = T(C) + 273.15.

T1 = 27°C + 273.15 = 300.15 K
T2 = 154°C + 273.15 = 427.15 K

Now we have the values:
V1 = 5.70 L
T1 = 300.15 K
V2 = ?
T2 = 427.15 K

Using Charles's law equation, we can write:

V1 / T1 = V2 / T2

Substituting the known values:

(5.70 L) / (300.15 K) = V2 / (427.15 K)

Solving for V2:

V2 = (5.70 L * 427.15 K) / (300.15 K)
V2 ≈ 8.13 L

Therefore, if the temperature is increased to 154°C while the pressure is held constant, the gas will occupy a volume of approximately 8.13 L.