 # Black principle – problems and solutions

Black principle – problems and solutions

1. 200-gram water at 30°C mixed with 100-gram water at 90°C. The specific heat of water = 1 cal.gram−1°C−1. Determine the final temperature of the mixture!

Known :

Mass of water 2 (m1) = 200 gram

Temperature of water 1 (T1) = 30oC

Mass of water 2 (m2) = 100 gram

Temperature of water 2 (T2) = 90oC

The specific heat for water (c) = 1 cal.gram−1°C−1

Wanted : The final temperature

Solution :

Heat released by hooter water (Q2) = heat absorbed by cooler water (Q1)

m2 c (ΔT) = m1 c (ΔT)

(100)(1)(90 – T) = (200)(1)(T – 30)

(100)(90 – T) = (200)(T – 30)

9000 – 100T = 200T – 6000

9000 + 6000 = 200T + 100T

15000 = 300T

T = 15000/300

T = 50oC

2. 60-gram at 90oC mixed with 40-gram water at 25oC. What is the final temperature of the mixture. The specific heat for water = 1 cal.g-1.oC-1.

Known :

Mass of water 1 (m1) = 60 gram

Temperature of water 1 (T1) = 90oC

Mass of water 2 (m2) = 40 gram

Temperature of water 2 (T2) = 25oC

The specific heat for water (c) = 1 cal.g-1.oC-1

Wanted : The final temperature

Solution :

Heat released by hooter water (Q2) = heat absorbed by cooler water (Q1)

m1 c (ΔT) = m2 c (ΔT)

(60)(1)(90 – T) = (40)(1)(T – 25)

(60)(90 – T) = (40)(T – 25)

5400 – 60T = 40T – 1000

5400 + 1000 = 40T + 60T

6400 = 100T

T = 6400/100

T = 64oC

3. M-gram ice at 0oC placed in 340-gram water at 20oC in a specific container. Latent heat of fusion (Lice) = 80 cal g-1, the specific heat of water (cwater) = 1 cal g-1 oC-1. All the ice melts and the thermal equilibrium = 5oC. Find mass of ice.

Known :

Mass of water (m) = 340 gram

Temperature of ice (Tice) = 0oC

Temperature of water (Twater) = 20oC

Temperature of thermal equilibrium (T) = 5oC

Latent heat of fusion for ice (Lice) = 80 cal g-1

The specific heat for water (cwater) = 1 cal g-1 oC-1

Wanted : Mass of ice (M)

Solution :

Heat released by water (Q2) = heat absorbed by ice (Q1)

mwater cwater (ΔT) = mice Lice + mice cwater (ΔT)

(340)(1)(20-5) = M (80) + M (1)(5-0)

(340)(15) = 80M + 5M

5100 = 85M

M = 5100/85

M = 60 gram

4. A copper at 100oC placed in 128-gram of water at 30 oC. The specific heat for water is 1 cal.g-1oC-1 and the specific heat for copper is 0.1 cal.g-1oC-1. If the temperature of the thermal equilibrium = 36 oC, what is the mass of copper.

See also  Centripetal force in uniform circular motion – problems and solutions

Known :

Temperature of copper (T1) = 100 oC

The specific heat for copper (c1) = 0.1 cal.g-1oC-1

Mass of water (m2) = 128 gram

Temperature of water (T2) = 30 oC

The specific heat for water (c2) = 1 cal.g-1oC-1

The temperature of the thermal equilibrium (T) = 36 oC

Wanted : Mass of copper (m1)

Solution :

Q copper = Q water

m1 c1 ΔT = m2 c2 ΔT

(m1)(0.1)(100-36) = (128)(1)(36-30)

(m1)(0.1)(64) = (128)(1)(6)

(m1)(6.4) = 768

m1 = 768 / 6.4

m1 = 120 gram

5. A 3-kg lead with the specific heat for lead is 1400 J.kg-1C-1 at 80oC placed in 10-kg water with the specific heat for water is 4200 J.kg-1C-1. The temperature of the thermal equilibrium is 20oC. What is the initial temperature of water.

Known :

Mass of lead (m1) = 3 kg

The specific heat for lead (c1) = 1400 J.kg-1C-1

Temperature of lead (T1) = 80 oC

Mass of water (m2) = 10 kg

The specific heat of water (c2) = 4200 J.kg-1C-1

The temperature of the thermal equilibrium (T) = 20 oC

Wanted : The initial temperature of water (T2)

Solution :

Q release = Q absorb

m1 c1 ΔT = m2 c2 ΔT

(3)(1400)(80-20) = (10)(4200)(20-T)

(4200)(60) = (42,000)(20-T)

252,000 = 840,000 – 42,000 T

42,000 T = 840,000 – 252,000

42,000 T = 588,000

T = 588,000 / 42,000

T = 14

The initial temperature of water is 14oC.

6. 75-gram water at 0oC mixed with 50-gram water so the temperature of mixture is 40oC. What is the initial temperature of 50-gram water.

Known :

Mass of water 1 (m1) = 75 gram

The initial temperature of water 1 (T1) = 0oC

Mass of water 2 (m2) = 50 gram

The temperature of mixture (T) = 40oC

Wanted : The initial temperature of water 2 (T2)

Solution :

Heat released by hooter water (Qrelease) = heat absorbed by cooler water (Qabsorbs)

m1 c (ΔT1) = m2 c (ΔT2)

m1 (ΔT1) = m2 (ΔT2)

(75)(40 – 0) = (50)(T2 – 40)

(75)(40) = (50)(T2 – 40)

3000 = 50 T2 – 2000

3000 + 2000 = 50 T2

5000 = 50 T2

T2 = 100 oC

7. 200-gram metal heated to 120oC, then placed in 100-gram water at 30oC. The temperature of mixture at thermal equilibrium is 60oC. If the specific heat of water is 4200 J.kg-1 oC-1, what is the specific heat of the metal.

See also  The energy produced in the fusion reaction – problems and solutions

Solution :

Convert mass unit from gram to kilogram (International unit)

Known :

Mass of metal (m1) = 200 gram = 0.2 kg

Temperature of metal (T1) = 120oC

Mass of water (m2) = 100 gram = 0.1 kg

Temperature of water (T2) = 30oC

The temperature of mixture (T) = 60oC

The specific heat of water (c2) = 4200 J.kg-1 oC-1

Wanted : The specific heat of metal (c1)

Solution :

Qrelease = Qabsorb

m1 c1 (ΔT1) = m2 c2 (ΔT2)

(0.2)(c1)(120 – 60) = (0.1)(4200)(60 – 30)

(0.2)(c1)(60) = (0.1)(4200)(30)

12 c1 = 12600

c1 = 12600 / 12

c1 = 1050

1. Question: Who was Joseph Black? Answer: Joseph Black was an 18th-century Scottish physician and chemist, known for his discoveries of latent heat, specific heat, and carbon dioxide.

2. Question: What is latent heat? Answer: Latent heat is the amount of energy absorbed or released by a substance during a phase change, like melting or boiling, without changing its temperature.

3. Question: How does latent heat differ from sensible heat? Answer: While sensible heat causes a change in temperature of a substance without a phase change, latent heat causes a phase change without a change in temperature.

4. Question: What significance did Black’s experiments on latent heat have on the study of thermodynamics? Answer: Black’s work laid the foundational understanding of heat energy and its conservation, paving the way for further advancements in thermodynamics and calorimetry.

5. Question: Why does ice at 0°C and water at 0°C have different amounts of heat energy? Answer: Ice at 0°C needs to absorb the latent heat of fusion to become water at 0°C. Thus, water at 0°C has more heat energy than ice at the same temperature due to this absorbed latent heat.

6. Question: What is specific heat? Answer: Specific heat is the amount of heat energy required to raise the temperature of a unit mass of a substance by one degree Celsius or Kelvin.

7. Question: How did Black’s work on specific heat lead to the understanding of different heat capacities of substances? Answer: Black discovered that equal masses of different substances required different amounts of heat to achieve the same change in temperature. This observation introduced the concept of specific heat capacities of substances.