Question

A 24-cm-diameter vertical cylinder is sealed at the top by a frictionless 15 kg piston. The piston is 90 cm above the bottom when the gas temperature is 315 ∘C. The air above the piston is at 1.00 atm pressure.
A) What is the gas pressure inside the cylinder?B) What will the height of the piston be if the temperature is lowered to18 ∘C?

Answer 1

Answer:

(A) $P_i=3249.41\ Pa$

(B) $V_f=0.3358\ m$

Explanation:

Given:

• diameter of the cylinder, $d= 0.24\ m$

• mass of piston sealing on the top, $m_p=15\ kg$

• initial temperature of the piston, $T_i=315+273= 588\ K$

• initial height of piston, $h_i=0.9\ m$

• atmospheric pressure on the piston, $p_a=1 atm=101325\ Pa$

(A)

Initial pressure of gas is the pressure balanced by the weight of piston:

$P_i=\frac{m_p.g}{\pi.d^2\div 4}$

$P_i=\frac{15\times 9.8}{\pi\times (0.24^2\div 4)}$

$P_i=3249.41\ Pa$

Which is gauge pressure because it is measured with respect to the atmospheric pressure.

(B)

Given:

• Final temperature, $T_f=18+273=291\ K$

Now, volume of air initially in the cylinder:

$V_i=\pi.d.h_i$

$V_i=\pi\times 0.24\times 0.9$

$V_i=0.6786\ m^3$

Using gas law:

$\frac{P_i V_i}{T_i}= \frac{P_f V_f}{T_f}$ ........................................(1)

∵In every condition of equilibrium the gas pressure will be balanced by the weight of the piston so it is an isobaric transition.

∴$P_i=P_f$

Hence eq. (1) is reduced to:

$\frac{V_i}{T_i}= \frac{V_f}{T_f}$

putting respective values:

$\frac{0.6786}{588}= \frac{V_f}{291}$

$V_f=0.3358\ m$

Answer 2

Considering the gasses law and all the parameters, 1) the pressure inside the cylinder is 3249.4 Pa, 2) the height of the piston at 18ºC is 0.4452 m.

What is the Law of gasses?

The law of gasses is a group of chemical and physical laws that allows understanding the behavior of gasses in a close system.

The law of gasses considers different parameters, such as

• Pressure, P: the amount of force applied on a surface. It is expressed in Pascals (Pa) or atmospheres (atm). 1 atm = 101325 Pa.

• Volume, V: spaces occupied by a certain amount of mass. It is expressed in litters (L).

• Temperature, T: it is a measure of the particles' internal agitation. It is expressed in kelvins (K). One centigrade = 1 + 273 kelvins.

When talking about standard conditions in gas, we refer to 1 atm pressure, 273 K of temperature (0ºC), and 22.4 L/mol of volume.

The different laws are,

• Boyle law ⇒ P₁V₁ = P₂V₂ ⇒ P and V are inversely proportional. T is constant.

• Charles law ⇒ V₁/T₁ = V₂/T₂ ⇒ V and T are directily porportional. P is constant.

• Gay-Lussac law ⇒ P₁/T₁ = P₂/T₂ ⇒ T and P are directily porportional. V is constant.

The relationship between these three laws and the variables is as follows,

                                 (P₁V₁) / T₁ = (P₂V₂) / T₂

According to this framework, we can answer the questions in this porblem.

Available data:

• D = 24-cm² = 0.0024 m²
• m = 15 kg
• h = 90 cm = 0.9 m
• T₀ = 315ºC ⇒ 315 + 273 = 588 K.
• Pair = 1atm

A) What is the gas pressure inside the cylinder?

We know that P is defined by the relationship between force and area.

P = F/A

So, let us define Force and Area.

• FORCE

F is the force applied by the piston. So we need to get the F of the piston.

To do it, we will use the following formula,

                             

 W = mg             ⇒  Where m is mass (15 kg) and g is gravity (9.8m/s²)

Since W is a type of force, we can replace this value in the previous general formula,

P = F/A = W/A = mg/A

• AREA

Area A is defined by the following formula

A = π r²

Where

- π = 3.1416

- r = D/2

Now, we can replace A in the general formula as follows,

P = mg/A = mg / π (D/2)²

The pressure inside the cylinder is,

P = mg / π (D/2)²

P = (15 x 9.8) / 3.1416 (12)²

P = 147 / 452.39

P = 0.32494

P = 3249.4 Pa

The gas pressure inside the cylinder 3249.4 Pa.

B) What will the height of the piston be if the temperature is lowered to 18 ∘C?

T₁ = 588 K

T₂ = 18 ºC ⇒ 18 + 273 = 291 K

Now, volume can be calculated as follows,

V = π r² h = A h

Where

- π = 3.1416

- r = D/2 = 12

- h₁ = 0.9 m

- A = 452.39 cm² = 0.4524 m²

Replacing,

V₁ = A h₁

V₁ = 0.4524 m² x  0.9 m

V₁ = 0.4071 m³

Now, we have the initial volume V₁, and the inicial and final temperatures, T₁ and T₂. We will use this values and the Charles law to get the final volume, V₂.

V₁/T₁ = V₂/T₂

V₂ = T₂ (V₁/T₁)

V₂ = 291 (0.4071/588)

V₂ = 291 (0.4071/588)

V₂ =0.2014

Now that we have the final volume, we can use this value to get the final height.

V₂ = A h₂

h = V₂/A

h₂ = 0.2014 m³ / 0.4524 m²

h₂ = 0.4452 m

The height of the piston when the temperature is lowered to 18ºC is 0.4452 m.

You can learn more about gasses law at

https://brainly.com/question/12807582

https://brainly.com/question/1437490

https://brainly.com/question/3491421

https://brainly.com/question/21184611

https://brainly.com/question/2683502