Question

The cylinder in the figure(Figure 1) has a moveable piston attached to a spring. The cylinder's cross-section area is 10 cm2, it contains 0.0048 mol of gas, and the spring constant is 1500 N/m. At 21 ∘C the spring is neither compressed nor stretched. How far is the spring compressed if the gas temperature is raised to 200 ∘C? Express your answer to two significant figures and include the appropriate units.

Answer 1

Hi,

Hope you are doing well.

We know that, Gases expand as temperature increases. Here, when temperature increases from 21^oC to 200^oC, gas inside the cylinder will expand and the piston will compress the spring attached to the piston.

lets assume that the gas inside the cylinder is an ideal gas.

We know that,

Work done,

where,

'' is the force acting on piston/spring.(N)

'' is the displacement of piston/spring.(m)

let '' be the change in volume of gas (m³) and '' be the area of cross-section of cylinder(m²).

where,

'' is the displacement of piston/spring.(m)

also,

Where,

'' is the pressure of gas.(N/m²)

Multiplying both sides we get,

Where,

'' is the Work done by gas to move piston/ work done by spring(J).

'' is the pressure of gas.(N/m²)

'' is the change in volume of gas (m³).

Also, we know that,

For an ideal gas,

Where,

'' is the pressure of gas.(N/m²)

'' is the Volume of gas.(m³)

'' is the no.of moles of gas.

'' is the Universal Gas Constant= 8.314 J/mol-K

'' is the temperature of gas.(K)

Differentiating both sides, we get,

We know that work done by spring due to compression,

Where,

'' is the spring constant.(N/m)

'' is the displacement of piston/spring.(m)

WORK DONE BY GAS = WORK DONE TO COMPRESS SPRING

Given that,

Spring constant,

No.of moles of gas,

Universal Gas Constant,

Initial temperature,

Final temperature,

Change in temperature,

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