We mix 50.0 mL of ethanol (density = 0.789 g>mL) initially at 7.0 C with 50.0 mL of water (density = 1.0 g>mL) initially at 28.4 C in an insulated beaker. Assuming that no heat is lost, what is the final temperature of the mixture?

Understanding how to calculate the final temperature of a mixture involves using the heat exchange equation. This indispensable tool in thermodynamics is based on the premise that heat transferred out of one substance is equal to the heat absorbed by another, when they are placed in contact within an isolated system.

The equation takes this general form: \( Q_{lost} = Q_{gained} \), where \( Q \) represents the heat quantity. When dealing with mixtures, you can express this for each substance in the mixture as \( m \times c \times \Delta T \), where:\

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• \m\ represents the mass of the substance,\
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• \c\ is the specific heat capacity,\
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• \( \Delta T \) is the change in temperature.\
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For instance, when mixing ethanol and water, the equation helps us set up a system of equations where the heat given off by the water as it cools is precisely equal to that absorbed by the ethanol as it warms up.

The specific heat capacity (\( c \)) is a property of a substance that indicates how much heat is needed to raise the temperature of one gram of the substance by one degree Celsius (or Kelvin). Different substances have different specific heat capacities, which is why they warm up or cool down at varying rates.

For example, in our mixture, ethanol has a specific heat capacity of 2.44 J/g°C, and water has a higher specific heat capacity of 4.18 J/g°C. This means that per gram, water requires more heat energy to undergo the same temperature change as ethanol. Hence, water will warm up more slowly and cool down more slowly compared to ethanol, affecting the final temperature of the mixture.

A cornerstone of thermodynamics applied in mixing substances is the principle that 'heat gained is equal to heat lost' within a closed system. This principle ensures that all the heat coming from a warmer substance is accounted for by the cooler substance absorbing it, given no heat loss to the surroundings.

In practical terms, when calculating the final temperature of a mixed substance, this concept ensures that: \( m_{hot} \times c_{hot} \times (T_{initial,hot} - T_{final}) = m_{cold} \times c_{cold} \times (T_{final} - T_{initial,cold}) \). This means that the heat lost by the warmer substance (like water in our example) will be gained by the cooler substance (the ethanol). Applying this allows us to solve directly for the final temperature that balances these two heat exchanges.

Energy conservation is a fundamental principle stating that energy cannot be created or destroyed; it only changes form. In the context of mixtures, when two substances with different temperatures are mixed, the conservation of energy implies that the total energy before mixing is the same as the total energy after reaching equilibrium.

In our ethanol-water mix, energy in the form of heat is moved from the water to the ethanol until they reach a common temperature. No energy is lost to the surroundings due to the insulated beaker. By using the mass and specific heat capacity of each, along with their respective temperature changes, we can apply the conservation of energy to find the final unified temperature of the mixture.