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10 Heat and Matter
10.13 Y =
F/A
L/L
(1)
or F =
Y AL
L
=
Y AL
L 0 (1 + ααT )
Y AL
L 0
(2)
(∵ ααT << 1)
But L = α L 0 T
∴ F = Y AααT = 2.1 × 10
10
× 0.5 × 10
−6
× 12 × 10
−6
× 20 = 2.52 N
where we have used SI units.
10.14 The coefficient of apparent expansion of a liquid
A = γ − g = γ − 3α = 11 × 10
−4
− 3 × 8 × 10
−6
= 1.076 × 10
−3
Apparent expansion =
mass expelled
(mass left) (temperature rise)
A =
W
W 0 T
=
50 − W 0
W 0 T
∴ W 0 =
50
1 + AT
=
50
1 + 1.076 × 10 −3 × 80
= 46 g
Gas Laws
10.15 PV = n RT (gas equation)
∴ R =
PV
nT
=
(1.0129 × 10 5 ) (22.4 × 10 −3 )
(1.0) (273)
= 8.31 J/mol/K
10.16 P 0 = 0.76 × 13, 600 × 9.8 = 1.0129 × 10 5 Pa
Pressure at depth 30 m, P = 30 × 1000 × 9.8 = 2.94 × 10 5 Pa.
∴ Total pressure inside the bubble, P 1 = P 0 + P
= (1.0129 + 2.94) × 10
5
= 3.9529 × 10
5 Pa
P 2 V 2 = P 1 V 1
(Boyle’s law)
∴ P 2 (3V 1 ) = P 1 V 1
∴ P 2 = P 1 /3 = 1.376 × 10
5 Pa
This corresponds to a water depth equivalent of 1.3176 × 10 5 − 1.0129 ×
10 5 = 0.3047 × 10 5 Pa.
Therefore water depth =
30 × 0.3047 × 10 5
2.94 × 10 5
= 3.11 m