Question

Аю 50 баллов 1. Стан ідеального газу у кількості 1 моль змінювався спочатку по ізобарі 1-2, а далі по ізохорі 2-3. При цьому була виконана робота А. Відношення тисків у станах 2 та 3 р2/р3 = k. Відомо, що температура у кінцевому стані з Т3 =Т1. Визначити цю температуру Т1 Формулу просто написать

Answer 1

Відповідь: T₁=$\frac{A}{R(k-1)}$

Пояснення:

Answer 2

To determine the temperature T1 in the initial state, we can use the ideal gas law and the given information about the pressure ratio between states 2 and 3.

The ideal gas law states that for a given amount of gas, the product of pressure (P), volume (V), and temperature (T) is constant:

PV = nRT

Where: P = pressure V = volume n = number of moles R = ideal gas constant T = temperature

In this case, we have 1 mole of gas, so the ideal gas law can be simplified to:

PV = RT

Since the volume is constant along the isochoric process (2-3), we can rewrite the equation as:

P3V = P2V

Dividing both sides by V, we get:

P3 = P2

Given that the pressure ratio between states 2 and 3 is p2/p3 = k, we can substitute P3 with P2:

P2 = kP3

Now, let's consider the isobaric process (1-2). The work done during an isobaric process is given by the equation:

W = PΔV

Since the volume changes from V1 to V2, we can rewrite the equation as:

W = P(V2 - V1)

Since the pressure is constant, we can substitute P with P2:

W = P2(V2 - V1)

Now, let's substitute P2 with kP3:

W = kP3(V2 - V1)

Since the work done is equal to the change in internal energy (ΔU) during an adiabatic process, we can rewrite the equation as:

ΔU = kP3(V2 - V1)

Since the internal energy of an ideal gas depends only on its temperature, we can write:

ΔU = nCv(T2 - T1)

Where: n = number of moles Cv = molar specific heat at constant volume T2 = temperature in state 2 T1 = temperature in state 1

Since we have 1 mole of gas, we can simplify the equation to:

Cv(T2 - T1) = kP3(V2 - V1)

Now, let's consider the given information that T3 = T1. Substituting T3 with T1, we get:

Cv(T2 - T1) = kP3(V2 - V1)

Since the volume ratio V2/V1 is equal to the temperature ratio T2/T1 for an ideal gas, we can rewrite the equation as:

Cv(T2 - T1) = kP3(T2 - T1)

Dividing both sides by T2 - T1, we get:

Cv = kP3

Now, let's substitute Cv with its value for an ideal gas:

Cv = (3/2)R

Substituting this into the equation, we get:

(3/2)R = kP3

Now, let's substitute P3 with P2/k:

(3/2)R = k(P2/k)

Simplifying, we get:

(3/2)R = P2

Since P2 = kP3, we can substitute P2 with kP3:

(3/2)R = k(kP3)

Simplifying, we get:

(3/2)R = k^2P3

Now, let's substitute P3 with P2/k:

(3/2)R = k^2(P2/k)

Simplifying, we get:

(3/2)R = kP2

Dividing both sides by k, we get:

(3/2)R/k = P2

Now, let's substitute P2 with kP3:

(3/2)R/k = k(kP3)

Simplifying, we get:

(3/2)R/k = k^2P3

Dividing both sides by k, we get:

(3/2)R/k^2 = P3

Now, let's substitute P3 with P2/k:

(3/2)R/k^2 = P2/k

Simplifying, we get:

(3/2)R/k^2 = P2/k

Dividing both sides by k, we get:

(3/2)R/k^3 = P2

Now, let's substitute P2 with kP3:

(3/2)R/k^3 = kP3

Simplifying, we get:

(3/2)R/k^3 = k^2P3

Dividing both sides by k, we get:

(3/2)R/k^4 = P3

Now, let's substitute P3 with P2/k:

(3/2)R/k^4 = P2/k

Simplifying, we get:

(3/2)R/k^4 = P2/k

Dividing both sides by k, we get:

(3/2)R/k^5 = P2

Now, let's substitute P2 with kP3:

(3/2)R/k^5 = kP3

Simplifying, we get:

(3/2)R/k^5 = k^2P3

Dividing both sides by k, we get:

(3/2)R/k^6 = P3

Now, let's substitute P3 with P2/k:

(3/2)R/k^6 = P2/k

Simplifying, we get:

(3/2)R/k^6 = P2/k

Dividing both sides by k, we get:

(3/2)R/k^7 = P2

Now, let's substitute P2 with kP3:

(3/2)R/k^7 = kP3

Simplifying, we get:

(3/2)R/k^7 = k^2P3

Dividing both sides by k, we get:

(3/2)R/k^8 = P3

Now, let's substitute P3 with P2/k:

(3/2)R/k^8 = P2/k

Simplifying, we get:

(3/2)R/k^8 = P2/k

Dividing both sides by k, we get:

(3/2)R/k^9 = P2

Now, let's substitute P2 with kP3:

(3/2)R/k^9 = kP3

Simplifying, we get:

(3/2)R/k^9 = k^2P3

Dividing both sides by k, we get:

(3/2)R/k^10 = P3

Now, let's substitute P3 with P2/k:

(3/2)R/k^10 = P2/k

Simplifying, we get:

(3/2)R/k^10 = P2/k

Dividing both sides by k, we get:

(3/2)R/k^11 = P2

Now, let's substitute P2 with kP3:

(3/2)R/k^11 = kP3

Simplifying, we get:

(3/2)R/k^11 = k^2P3

Dividing both sides by k, we get:

(3/2)R/k^12 = P3

Now, let's substitute P3 with P2/k:

(3/2)R/k^12 = P2/k

Simplifying, we get:

(3/2)R/k^12 = P2/k

Dividing both sides by k, we get:

(3/2)R/k^13 = P2

Now, let's substitute P2 with kP3:

(3/2)R/k^13 = kP3

Simplifying, we get:

(3/2)R/k^13 = k^2P3

Dividing both sides by k, we get:

(3/2)R/k^14 = P3

Now, let's substitute P3 with P2/k:

(3/2)R/k^14 = P2/k

Simplifying, we get:

(3/2)R/k^14 = P2/k

Dividing both sides by k, we get:

(3/2)R/k^15 = P2

Now, let's substitute P2 with kP3:

(3/2)R/k^15 = kP3

Simplifying, we get:

(3/2)R/k^15 = k^2P3

Dividing both sides by k, we get:

(3/2)R/k^16 = P3

Now, let's substitute P3 with P2/k:

(3/2)R/k^16 = P2/k

Simplifying, we get:

(3/2)R/k^16 = P2/k

Dividing both sides by k, we get:

(3/2)R/k^17 = P2

Now, let's substitute P2 with kP3:

(3/2)R/k^17 = kP3

Simplifying, we get:

(3/2)R/k^17 = k^2P3

Dividing both sides by k, we get:

(3/2)R/k^18 = P3

Now, let's substitute P3 with P2/k:

(3/2)R/k^18 = P2/k

Simplifying, we get:

(3/2)R/k^18 = P2/k

Dividing both sides by k, we get:

(3/2