Heat Capacity + Vaccum + Springs +Heat!

Classical Mechanics Level 5

The figure shows a system with adiabatic walls, a piston and a spring. The space on the left side of the piston is filled with 1 mole of an ideal mono atomic gas and right side is a vacuum with a spring. Initially the system is in equilibrium. Now the gas is heated. Molar heat capacity of the gas in this case is $C=\frac{nR}{2}$ . What is the value of $n$ is?

(Neglect heat capacities of wall, piston & spring)

The answer is 4.

1 solution

Nishu Sharma
May 14, 2015

Since Piston is masless , hence Forces are balanced on each side of it .

$\displaystyle{{ P }_{ g }A\quad \propto \quad Kx\\ { P }_{ g }{ A }^{ 2 }\quad \propto \quad KAx\\ { P }_{ g }{ A }^{ 2 }\quad \propto \quad KV\\ { P }_{ g }{ V }^{ -1 }\quad =\quad \lambda \\ \Rightarrow \quad x=-1\quad \\ C={ C }_{ v }+\cfrac { R }{ 1-x } \\ \boxed { C=\cfrac { 4R }{ 2 } } }$

This problem is not correct. Who told you that $Ax=V$ ?

Raghav Vaidyanathan - 6 years, 1 month ago

$kx=0$ when $P=0$ i.e no gas is injected . Hence $x$ is related to volume as $Ax=V$ .

Mahimn Bhatt - 6 years ago

Volume of the gas is $Al$ where $l$ is the length on the left hand side of piston. Agreed? Now how is $l=x$ ? It is not mentioned that the spring is as long as the container.

Raghav Vaidyanathan - 6 years ago

@Raghav Vaidyanathan – If we balance forces then force by gas=PA (Initial state)(Left hand side)

Then on the right hand side we have no force as spring is in uncompressed state.

Hence forces are not balanced.

Thus we are left with 1 case.(which is that the pressure due to gas=0)

Do reply.

Mahimn Bhatt - 6 years ago

@Mahimn Bhatt – Nope, pressure due to gas is not zero. It is given that the spring is initially in compressed state and the system is in equilibrium.

Raghav Vaidyanathan - 6 years ago

@Raghav Vaidyanathan – ohk got it

Mahimn Bhatt - 6 years ago

@Mahimn Bhatt – But basically we need to find out the relation between P and V. That way thinking makes it more easier to solve. Isnt it?

Md Zuhair - 3 years, 4 months ago

@Raghav Vaidyanathan – Yep thats an error.

In fact this is the third time this problem is posted on brilliant.

Kushal Patankar - 6 years ago

Heat Capacity + Vaccum + Springs +Heat!

The answer is 4.

1 solution

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