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Newton's Law of Cooling Calculus, Example Problems, Differential Equations

The Organic Chemistry Tutor・14 minutes read

The video focuses on Newton's law of cooling, demonstrating how to calculate temperature at any time using the formula 'ts + c * e^(-kt)' and determining the time it takes for an object to cool from 100°C to 70°C. The concept of temperature change rate being proportional to the temperature difference is emphasized, with the formula for temperature change rate derived and explained in detail.

Insights

  • The video focuses on Newton's law of cooling, which involves calculating temperature changes over time in an object based on its initial temperature, surrounding temperature, and a constant 'k'.
  • The formula derived from Newton's law of cooling, 'ts + c * e^(-kt)', allows for the calculation of temperature at any time 't' and demonstrates the relationship between temperature, time, and surrounding temperature, providing a comprehensive understanding of the cooling process.

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Recent questions

  • What is Newton's law of cooling?

    Newton's law of cooling describes how the temperature of an object changes over time when it is in contact with a cooler surrounding environment.

  • How is the constant 'k' determined in Newton's law of cooling?

    The constant 'k' in Newton's law of cooling is determined by using the initial temperature of the object, the surrounding temperature, and the time it takes for the object to cool from one temperature to another.

  • What is the formula for temperature change rate in Newton's law of cooling?

    The formula for temperature change rate in Newton's law of cooling shows the relationship between temperature, time, and the surrounding temperature, indicating that the rate of temperature change is proportional to the difference in temperatures.

  • How is the final formula for temperature at any time derived in Newton's law of cooling?

    The final formula for temperature at any time in Newton's law of cooling is derived by integrating and solving for the constant 'c' using the initial temperature of the object, resulting in the equation 'ts + c * e^(-kt)'.

  • How is the formula for Newton's law of cooling used to calculate cooling time?

    The formula for Newton's law of cooling can be used to calculate the time it takes for an object to cool from one temperature to another by plugging in the relevant values such as initial temperature, surrounding temperature, and the constant 'k', providing an accurate estimation of cooling time.

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Summary

00:00

Cooling Law: Calculating Temperature Change Over Time

  • Newton's law of cooling is the focus of the video.
  • An example problem is presented where an object cools from 100°C to 80°C in 12 minutes in a room at 50°C.
  • An equation is introduced to calculate temperature at any time, involving the object's initial temperature, surrounding temperature, and a constant 'k'.
  • The constant 'k' is determined by using the initial temperature, surrounding temperature, and the time taken for the object to cool from 100°C to 80°C.
  • The value of 'k' is found to be 0.04257.
  • The equation is then used to find the time it takes for the object's temperature to decrease from 80°C to 70°C, which is calculated to be 9.5 minutes.
  • The concept of temperature change rate being proportional to the difference in temperatures is explained.
  • The formula for temperature change rate is derived, showing the relationship between temperature, time, and the surrounding temperature.
  • The formula is further simplified by integrating and solving for the constant 'c' using the initial temperature.
  • The final formula for temperature at any time 't' is presented as 'ts + c * e^(-kt)'.
  • The process of using the formula to calculate the time it takes for the object to cool from 100°C to 70°C is demonstrated, resulting in 21.52 minutes.

22:12

Derivation of Newton's Law of Cooling Formula

  • Initial minus surrounding temperature equals constant c
  • Formula for Newton's Law of Cooling: t of t = surrounding temperature + difference between initial temperature and surrounding temperature times e raised to the negative kt
  • Derivation of the formula for Newton's Law of Cooling explained
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