Problem 87 - Chemistry and Chemical Reactivity

Question

Solve the following equation for the unknown value, \(T\) \((4.184)(244)(T-292.0)+(0.449)(88.5)(T-369.0)=0\)

Step-by-Step Solution

Answer

The value of T is approximately 294.669.

Step 1: Distribute Terms

First, distribute the constants across the terms in the parentheses for both parts of the equation. This means multiplying 4.184 by 244 and (T - 292.0), as well as multiplying 0.449 by 88.5 and (T - 369.0).

Step 2: Distribution Results

This gives: \[4.184 \times 244 \times T - 4.184 \times 244 \times 292.0 + 0.449 \times 88.5 \times T - 0.449 \times 88.5 \times 369.0 = 0\].Perform the multiplications: 1. 4.184 × 244 = 1021.0562. 4.184 × 244 × 292 = 298136.352Thus, your equation becomes: \[1021.056T - 298136.352 + 39.7365T - 14606.2455 = 0\].

Step 2: Combine Like Terms

Combine the terms involving T from the distribution.\[(1021.056 + 39.7365)T - 298136.352 - 14606.2455 = 0\].Calculate: 1. 1021.056 + 39.7365 = 1060.79252. 298136.352 + 14606.2455 = 312742.5975 This simplifies your equation to: \[1060.7925T - 312742.5975 = 0\].

Step 3: Solve for T

To isolate T, add 312742.5975 to both sides of the equation:\[1060.7925T = 312742.5975\].Then, divide both sides by 1060.7925:\[T = \frac{312742.5975}{1060.7925}\].Calculate the final result:\[T \approx 294.669\].

Key Concepts

Mathematical Distribution

Mathematical distribution is a fundamental concept in algebra that involves multiplying each term inside a parenthesis by a factor outside the parenthesis. It allows for the simplification of expressions, making it easier to manipulate and solve equations.
In the exercise provided, we have two distribution tasks involving multiplication of constants with terms inside parentheses. The idea is to ensure that every term inside the parentheses is multiplied appropriately to form a linear equation. For example:

• The expression \((4.184)(244)(T-292.0)\) becomes \(4.184 \times 244 \times T - 4.184 \times 244 \times 292.0 \).
• Similarly, \((0.449)(88.5)(T-369.0)\) becomes \(0.449 \times 88.5 \times T - 0.449 \times 88.5 \times 369.0 \).

Breaking down complex equations into distributed terms simplifies the next steps of equation solving. Remember, perform distribution first before any further algebraic operations.

Combining Like Terms

Combining like terms is the practice of merging terms in an expression that have the same variable or are constant. Doing so reduces complexity and simplifies the equation, making it easier to handle.
In the problem, we encounter terms that contain the unknown \(T\) and constant terms. Our aim is to put similar terms together to form groups, allowing us to work with simpler expressions.For instance:

• The terms \(1021.056T\) and \(39.7365T\) both involve \(T\). Combine them to form \((1021.056 + 39.7365)T = 1060.7925T\).
• The constant values \(-298136.352\) and \(-14606.2455\), when added, yield \(-312742.5975\).

By combining like terms, our equation becomes much more straightforward: \(1060.7925T - 312742.5975 = 0\). This new equation is easier to solve, as it focuses purely on two key parts - the term with \(T\) and the constant.

Solving for an Unknown

Solving for an unknown means rearranging an equation to isolate the variable on one side, providing its value. This is a critical skill in algebra for determining specific values that satisfy an equation.
Let's take the simplified equation \(1060.7925T - 312742.5975 = 0\). Our goal here is to "solve for \(T\)".
The steps involved are straightforward:

• Add \(312742.5975\) to both sides of the equation. This results in \(1060.7925T = 312742.5975\).
• To complete the isolation of \(T\), divide both sides by \(1060.7925\). This yields \(T = \frac{312742.5975}{1060.7925}\).
• Finally, compute the value to find that \(T \approx 294.669\).

These steps are crucial in solving equations, whether dealing with numbers or theoretical expressions involving algebraic symbols. Mastery of isolating and solving for unknowns enhances your algebraic proficiency.