Solutions manual for mechanics and thermodynamics

101

2.If the number of molecules in an ideal gas is doubled and the volume is doubled, by how much does the pressure change if the temperature is held constant ?

SOLUTION

The ideal gas law is

P V = NkT

If T is constant then

P / NV

If N is doubled and V is doubled then P does not change.

3.If the number of molecules in an ideal gas is doubled, and the absolute temperature is doubled and the pressure is halved, by how much does the volume change ?

(Absolute temperature is simply the temperature measured in Kelvin.)

SOLUTION

SOLUTION

Now let's verify this.

y(x) = x4

y(x + ¢x) = (x + ¢x)4

= x4 + 4x3¢x + 6x2(¢x)2 + 4x(¢x)3 + (¢x)4

107

4. Prove that dxd (3x2) = 3dxdx2 .

SOLUTION

y(x) = 3x2

y(x + ¢x) = 3(x + ¢x)2 = 3x2 + 6x¢x + 3(¢x)2

Now take

y(x) = x2 ) dxdy = 2x

Thus

dxd (3x2) = 6x

= 3dxd x2

5.Prove that dxd (x + x2) = dxdx + dxdx2 .

SOLUTION

Take y(x) = x + x2

y(x + ¢x) = x + ¢x + (x + ¢x)2

=x + ¢x + x2 + 2x¢x + (¢x)2

dx dx2

dx

109

6.Verify the chain rule and product rule using some examples of your own.

SOLUTION

your own examples