0
You visited us 0 times! Enjoying our articles? Unlock Full Access!
Question

$$1\ mole$$ of a gas is changed from its initial state $$(15\ lit; 2\ atm)$$ to final state $$(4\ lit, 10\ atm)$$ reversinly. If this change can be represented by a straight line in $$p-V$$ curve, calculate maximum temperature, the gas attained.

Solution
Verified by Toppr

$$(V_1, P_1)=(15\ lit, 2\ atm)$$
$$(V_2, P_2)=(4\ lit, 10\ atm)$$

Eq. of line
$$P-P_1=\dfrac{P_1-P_2}{V_1-V_2}(V-V_1)$$

$$P=P_1+\dfrac{P_1-P_2}{V_1-V_2}(V-V_1)$$

$$\dfrac{nRT}{V}=P_1+\left( \dfrac{P_1-P_2}{V_1-V_2}\right)V-\left( \dfrac{P_1-P_2}{V_1-V_2}\right)V_1$$

$$T=\dfrac{1}{nR}\left\{ P_1V +\left( \dfrac{P_1-P_2}{V_1-V_2}\right)V^2-\left( \dfrac{P_1-P_2}{V_1-V_2}\right) V_1.V\right\}$$

For $$T$$ to be max $$\dfrac{dT}{dV}=0$$

$$\dfrac{dT}{dV}=\dfrac{1}{nR}\left\{ P_1+2\left( \dfrac{P_1-P_2}{V_1-V_2}\right)V-\left( \dfrac{P_1-P_2}{V_1-V_2}\right)V_1\right\}=0$$

$$2\left( \dfrac{P_1-P_2}{V_1-V_2}\right)V=\dfrac{P_1V_1 -P_2V_1}{V_1-V_2}-P_1=\dfrac{P_1V_1 -P_2V_1 -P_1V_1+P_1V_2}{V_1 -V_2}=\dfrac{P_1V_2-P_2V_1}{(V_1-V_2)}$$

$$V=\dfrac{P_1V_2-P_2V_1}{2(P_1-P_2)}=\dfrac{2\times 4-10\times 15}{2(2-10)}=8.875$$

$$T_{max}$$, for $$1\ mole$$, $$n=1$$

$$T_{max}=\dfrac{1}{R}\left\{ 2\times 8.875+\left( \dfrac{2-10}{15-4}\right)\times (8.875)^2-\left( \dfrac{2-10}{15-4}\right)\times 15\times 8.875 \right\}$$

$$=\dfrac{1}{R}\left\{ 17.75+\left( \dfrac{-8}{11}\right)\times 78.7656-\left( \dfrac{-8}{11}\right) \times 133.125 \right\}$$

$$=\dfrac{1}{R}\left\{ 17.75+96.82+(-57.28)\right\}=\dfrac{1}{R}\times 57.286=\dfrac{57.29}{0.0821}=697.80\ K$$

Was this answer helpful?
0
Similar Questions
Q1
$$1\ mole$$ of a gas is changed from its initial state $$(15\ lit; 2\ atm)$$ to final state $$(4\ lit, 10\ atm)$$ reversinly. If this change can be represented by a straight line in $$p-V$$ curve, calculate maximum temperature, the gas attained.
View Solution
Q2
$$1\ mole$$ of a gas is changed from its initial state $$(15\ lit, 2\ atm)$$ to final state $$(4\ lit, 10\ atm)$$ reversibly. If this change can be represented by a straight line in $$P-V$$ curve, calculate maximum temperature the gas attained.
View Solution
Q3
A mole of gas is changed from its initial state $$(15 \,L, 2 \,atm)$$ to final state $$(4L, 10 \,atm)$$ reversibly. If this change can be represented by a straight line in $$PV$$ curve, calculate maximum temperature the gas attained.
View Solution
Q4
1 mole of a gas is changed from its initial value (15L,2atm) to final state (4L,10atm) respectively. If the change can be represented by a straight line in P-V curve. Calculate the maximum temperature attained by the gas.
View Solution
Q5

One mole of a gas changed from its initial state (15L,2atm) to the final state (4L,10atm) reversibly. If this change can be represented by a straight line in PV curve. Maximum temperature (approximate), the gas attained is x×102K. Then the value of x is:

View Solution