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A person is found dead at 5 pm in a room whose temperature is 20 degrees Celsius. The body temperature is measured to be 25 degrees Celsius when found. The heat transfer coefficient is estimated to be 8 W/m2 K. Modeling the body as a 30 cm diameter, 1.7 cm long cylinder. Estimate the time of death of the person.
- a)13.55 seconds
- b)12.55 seconds
- c)11.55 seconds
- d)10.55 seconds
Correct answer is option 'D'. Can you explain this answer?
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A person is found dead at 5 pm in a room whose temperature is 20 degr...
Tt – T INFINITY / T i – T INFINITY = e – b T.
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A person is found dead at 5 pm in a room whose temperature is 20 degr...
Given data:
Temperature of the room = 20°C
Temperature of the body when found = 25°C
Heat transfer coefficient = 8 W/m²K
Diameter of the body = 30 cm
Length of the body = 1.7 cm
To find: Time of death of the person
Solution:
1. Calculate the surface area of the body:
The surface area of the cylinder can be calculated as:
Surface area = 2πrL + 2πr²
where r = radius of the cylinder = diameter / 2 = 15 cm
L = length of the cylinder = 1.7 cm
Substituting the values, we get:
Surface area = 2π(15 cm)(1.7 cm) + 2π(15 cm)²
Surface area = 319.98 cm²
2. Calculate the heat loss from the body:
The heat loss from the body can be calculated using Newton's law of cooling, which states that:
Rate of heat loss = hA(Tbody - Troom)
where h = heat transfer coefficient = 8 W/m²K
A = surface area of the body in m²
Tbody = temperature of the body when found = 25°C
Troom = temperature of the room = 20°C
Converting units, we get:
h = 8 W/m²K = 0.008 W/cm²K
A = 319.98 cm² = 0.031998 m²
Tbody = 25°C = 298 K
Troom = 20°C = 293 K
Substituting the values, we get:
Rate of heat loss = 0.008 W/cm²K × 0.031998 m² × (298 K - 293 K)
Rate of heat loss = 0.004031 W
3. Calculate the initial temperature of the body:
The initial temperature of the body can be calculated using the formula:
Tbody = Troom + (Tinitial - Troom)e^(-ht/mc)
where Tinitial = initial temperature of the body
t = time elapsed since death
m = mass of the body
c = specific heat capacity of the body
Assuming the mass of the body to be 70 kg (average for an adult), and the specific heat capacity of the body to be 3.5 J/gK, we get:
m = 70 kg × 1000 g/kg = 70000 g
c = 3.5 J/gK
Substituting the values, we get:
25°C = 20°C + (Tinitial - 20°C)e^(-0.008t/70000×3.5)
5 = (Tinitial - 20°C)e^(-0.0000008t)
ln(5) = ln(Tinitial - 20°C) - 0.0000008t
ln(Tinitial - 20°C) = ln(5) + 0.0000008t
Tinitial - 20°C = e^(ln(5) + 0.0000008t)
Tinitial = e^(ln(5) + 0.0000008t) + 20°C
4. Calculate the time of death:
Substituting the initial temperature of the body as 37°C (average human body temperature),
Temperature of the room = 20°C
Temperature of the body when found = 25°C
Heat transfer coefficient = 8 W/m²K
Diameter of the body = 30 cm
Length of the body = 1.7 cm
To find: Time of death of the person
Solution:
1. Calculate the surface area of the body:
The surface area of the cylinder can be calculated as:
Surface area = 2πrL + 2πr²
where r = radius of the cylinder = diameter / 2 = 15 cm
L = length of the cylinder = 1.7 cm
Substituting the values, we get:
Surface area = 2π(15 cm)(1.7 cm) + 2π(15 cm)²
Surface area = 319.98 cm²
2. Calculate the heat loss from the body:
The heat loss from the body can be calculated using Newton's law of cooling, which states that:
Rate of heat loss = hA(Tbody - Troom)
where h = heat transfer coefficient = 8 W/m²K
A = surface area of the body in m²
Tbody = temperature of the body when found = 25°C
Troom = temperature of the room = 20°C
Converting units, we get:
h = 8 W/m²K = 0.008 W/cm²K
A = 319.98 cm² = 0.031998 m²
Tbody = 25°C = 298 K
Troom = 20°C = 293 K
Substituting the values, we get:
Rate of heat loss = 0.008 W/cm²K × 0.031998 m² × (298 K - 293 K)
Rate of heat loss = 0.004031 W
3. Calculate the initial temperature of the body:
The initial temperature of the body can be calculated using the formula:
Tbody = Troom + (Tinitial - Troom)e^(-ht/mc)
where Tinitial = initial temperature of the body
t = time elapsed since death
m = mass of the body
c = specific heat capacity of the body
Assuming the mass of the body to be 70 kg (average for an adult), and the specific heat capacity of the body to be 3.5 J/gK, we get:
m = 70 kg × 1000 g/kg = 70000 g
c = 3.5 J/gK
Substituting the values, we get:
25°C = 20°C + (Tinitial - 20°C)e^(-0.008t/70000×3.5)
5 = (Tinitial - 20°C)e^(-0.0000008t)
ln(5) = ln(Tinitial - 20°C) - 0.0000008t
ln(Tinitial - 20°C) = ln(5) + 0.0000008t
Tinitial - 20°C = e^(ln(5) + 0.0000008t)
Tinitial = e^(ln(5) + 0.0000008t) + 20°C
4. Calculate the time of death:
Substituting the initial temperature of the body as 37°C (average human body temperature),
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A person is found dead at 5 pm in a room whose temperature is 20 degrees Celsius. The body temperature is measured to be 25 degrees Celsius when found. The heat transfer coefficient is estimated to be 8 W/m2 K. Modeling the body as a 30 cm diameter, 1.7 cm long cylinder. Estimate the time of death of the person.a)13.55 secondsb)12.55 secondsc)11.55 secondsd)10.55 secondsCorrect answer is option 'D'. Can you explain this answer?
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