In a 25 degree C room, hot coffee in a vacuum flask cools from 72 degree C to 54 degree in 8 hours.Find its temperature after another 8 hours.
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To find the temperature of the hot coffee after another 8 hours, we can use Newton's Law of Cooling. According to this law, the rate of change of temperature of an object is proportional to the difference in temperature between the object and its surroundings.
First, let's find the cooling constant, K. We can use the given information:
T1 = 72 degrees Celsius (initial temperature of the coffee)
T2 = 54 degrees Celsius (temperature after 8 hours)
t = 8 hours (time taken for the change in temperature)
The formula we will use is: T(t) = T(s) + (T(0) - T(s)) * e^(-Kt)
Where:
T(t) is the temperature after time t,
T(s) is the surrounding temperature,
T(0) is the initial temperature,
e is the base of the natural logarithm (approximately 2.71828),
K is the cooling constant.
Given that the surrounding temperature is 25 degrees Celsius, we can rearrange the formula to solve for K:
T(t) - T(s) = (T(0) - T(s)) * e^(-Kt)
T2 - T1 = (T1 - T(s)) * e^(-Kt)
54 - 72 = (72 - 25) * e^(-K * 8)
-18 = 47 * e^(-8K)
We can further simplify this equation by dividing both sides by 47:
-18/47 = e^(-8K)
Now, we can solve for K by taking the natural logarithm (ln) of both sides:
ln(-18/47) = -8K
K = ln(-18/47) / -8
Using a calculator, we find that K ≈ 0.0173.
Now that we have the cooling constant, we can determine the temperature after another 8 hours (16 hours in total). Let's denote it as T3:
T(16) = T(s) + (T(0) - T(s)) * e^(-K * 16)
T(16) = 25 + (72 - 25) * e^(-0.0173 * 16)
T(16) = 25 + 47 * e^(-0.277)
Using a calculator, we find that T3 ≈ 44.9 degrees Celsius.
Therefore, the temperature of the hot coffee after another 8 hours will be approximately 44.9 degrees Celsius.
To find the temperature of the hot coffee after another 8 hours, we can use Newton's Law of Cooling. This law states that the rate of change of temperature of an object is proportional to the difference between the object's temperature and the surrounding temperature.
The formula for Newton's Law of Cooling is: T(t) = Ta + (To - Ta) * exp(-kt)
Where:
- T(t) is the temperature of the object at time t
- Ta is the surrounding temperature (in this case, 25 degrees Celsius)
- To is the initial temperature of the object (72 degrees Celsius in this case)
- k is the cooling constant, which is dependent on various factors such as the material of the object and the nature of heat transfer
- exp(-kt) is the exponential function
We can find the cooling constant, k, using the information given. The coffee cools from 72 degrees Celsius to 54 degrees Celsius in 8 hours, so we can use these values to find k.
Using the formula T(t) = Ta + (To - Ta) * exp(-kt) and solving for k, we get:
k = -(1/t) * ln((T(t) - Ta) / (To - Ta))
Substituting the values:
k = -(1/8) * ln((54 - 25) / (72 - 25))
Calculating this, we find k ≈ -0.0755
Now, we can use the k value to find the temperature of the coffee after another 8 hours (t = 16) using the same formula:
T(t) = Ta + (To - Ta) * exp(-kt)
Substituting the values:
T(t) = 25 + (72 - 25) * exp(-0.0755 * 16)
Calculating this, we find T(t) ≈ 46.7 degrees Celsius.
Therefore, the temperature of the hot coffee after another 8 hours will be approximately 46.7 degrees Celsius.
Assume an exponential rate of decay.
T = To + (To - Ta)*ext(-kt)
where t is the time in hours. You could solve for k, but there is an easier way to do it.
In 8 hours, the initial 47 degrees above ambient temperature becomes 19 C. 40.4% of the in initial temperature difference remains after eight hours. In another 8 hours, that difference from ambient (Ta) will decrease to 0.404 * 19 = 7.7 degrees, or 32.7 degrees.