# Thermodynamics – calculator for solving mixing problems

An online calculator to solve thermodynamic equilibrium problems, such as finding the final temperature when mixing fluids, or finding the required temperature for one of the fluids to achieve a final mixed temperature

This online calculator can solve thermodynamic equilibrium problems, such as finding the final temperature when mixing fluids, or finding the required temperature for one of the fluids to achieve a final mixed temperature. The only condition is that there should not be any phase transition (or phase change) of substances. To solve the problem, it uses the thermal equilibrium equation, more on this below.

## Thermal equilibrium equation

In the process of reaching thermodynamic equilibrium, heat is transferred from the warmer to the cooler object. Two objects are in thermal equilibrium if no heat flows between them when they are connected by a path permeable to heat, that is, they both have the same temperature. This is called the zeroth law of thermodynamics. A system is said to be in thermal equilibrium with itself if the temperature within the system is spatially and temporally uniform.

The thermodynamic system is called a thermally isolated system if it does not exchange mass or heat energy with its environment. In physics, the law of conservation of energy states that the total energy of an isolated system in a given frame of reference remains constant — it is said to be conserved over time.

The first law of thermodynamics can be stated as follows: during an interaction between a system and its surroundings, the amount of energy gained by the system must be exactly equal to the amount of energy lost by the surroundings. In the case of a thermally isolated system, we can say that during an interaction between objects inside a system (until it reaches thermal equilibrium), the amount of energy gained by one object must be exactly equal to the amount of energy lost by another.

This is our thermal equilibrium equation.

In another form:
,
where n – number of objects in the system.

That is, the algebraic sum of all heat quantities (gained and lost) in a thermally isolated system equals zero.

If we replace heat quantities with the formula described here: Quantity of heat, we will get the following equation:

,
note that the final temperature for all substances (T1, T2, ... Tn) should be the same, because of thermal equilibrium.

This is the equation used by the calculator to find the unknown value. Also, the calculator can take into account the quantity of heat gained or lost to the surroundings. This allows a more broad range of problems to be solved.

To use the calculator, you need to correctly fill out the table describing interacting substances. The usage instructions for different scenarios are listed below the calculator. #### Mixed substances

SubstanceMass, kgSpecific heat, J/kg*CInitial temperature, CFinal temperature, C
Items per page:

You can use this field to provide known added ("flows in") heat energy (use minus sign, because it is consumed from the environment) or known removed ("flows out") heat energy (use plus sign, because it is supplied to the environment). Zero stands for thermally isolated system and ? (question mark) stands for unknown value to be computed.
Digits after the decimal point: 1
Unknown quantity

Solution

### Example problems

There are a set of problems that can be solved using this calculator: final temperature of mixed fluids; required temperature for one of the fluids to achieve final mixed temperature; required mass for one of the fluids to achieve final mixed temperature; unknown specific heat; required quantity of heat, etc.

Here is how to use this calculator for different kinds of problems.

#### Example 1

A 600g piece of silver at 85.0 C is placed into 400g of water at 17.0 C in a 200g brass calorimeter. The final temperature of the water in the calorimeter is 22.0 C. What is the specific heat of silver?

How to use the calculator:

1. Clear the table by pressing the button Clear table
Substance Mass, kg Specific heat, J/kg*C Initial temperature, C Final temperature, C
Brass 0.2 380 17 22
Water 0.4 4200 17 22
Silver 0.6 ? 85 22

Pay attention to the question mark in silver's specific heat cell

1. The calculator solves the problem and outputs the solution – specific heat : 232.3 J/(kg*C), which is quite close to the table value for the specific heat of silver.

#### Example 2

3kg of water at 20.0 C reaches boiling point in a 1kg aluminium vessel. The specific heat of the water is 4200 J/(kgC), the specific heat of the aluminium is 920 J/(kgC). What is the required quantity of heat?

How to use calculator:

1. Clear the table by pressing the button Clear table
Substance Mass, kg Specific heat, J/kg*C Initial temperature, C Final temperature, C
Water 3 4200 20 100
Aluminium 1 920 20 100
1. Input ? (question mark) in Heat field.
2. The calculator solves the problem and outputs the solution – heat : -1081600 Joules. Minus means that the surroundings lost this amount to boil the water.

#### Example 3

A 2kg piece of lead at 90.0 C is placed into 1kg of water at 20.0 C in a 100g copper calorimeter. What is the final temperature of the water (assuming there is no heat loss to the environment)?

How to use the calculator:

1. Clear the table by pressing the button Clear table
Substance Mass, kg Specific heat, J/kg*C Initial temperature, C Final temperature, C
Copper 0.1 390 20 ?
Water 1 4200 20 ?

Pay attention to the question marks in all final temperature cells

1. The calculator solves the problem and outputs the solution – final temperature : 24.0 C

#### Specific heats table

Sometimes a problem does not list specific heats for involved substances. Normally you can look them up in handbooks, but for ease I have listed some of them below.

Substance Specific heat, J/kg*C
Aluminum 880
Acetone 2180
Benzene 1700
Bismuth 130
Water 4200
Glycerin 2400
Germanium 310
Iron 457
Gold 130
Potassium 760
Brass 380
Lithium 4400
Magnesium 1300
Copper 390
Sodium 1300
Nickel 460
Tin 230
Mercury 138
Silver 235
Ethyl alcohol 2430
Steel 460
Cast iron 500

Sources:

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