# mno2+hcl=mncl2+cl2+h2o

To balance the equation MnO2 + HCl = MnCl2 + Cl2 + H2O using the algebraic method step-by-step, you must have experience solving systems of linear equations. The most common methods are substitution/elimination and linear algebra, but any similar method will work.

### Step 1: Label Each Compound With a Variable

Label each compound (reactant or product) in the equation with a variable lớn represent the unknown coefficients.

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a MnO2 + b HCl = c MnCl2 + d Cl2 + f H2O

### Step 2: Create a System of Equations

Create an equation for each element (Mn, O, H, Cl) where each term represents the number of atoms of the element in each reactant or product.

```Mn:	1a	+	0b	=	1c	+	0d	+	0f
O:	2a	+	0b	=	0c	+	0d	+	1f
H:	0a	+	1b	=	0c	+	0d	+	2f
Cl:	0a	+	1b	=	2c	+	2d	+	0f

```

### Step 3: Solve For All Variables

Use substitution, Gaussian elimination, or a calculator lớn solve for each variable.

• 1a - 1c = 0
• 2a - 1f = 0
• 1b - 2f = 0
• 1b - 2c - 2d = 0

Use your graphing calculator's rref() function (or an online rref calculator) lớn convert the following matrix into reduced row-echelon-form:

Xem thêm: chuyển hỗn số thành số thập

```[ 1	 0	-1	 0	 0	0]
[ 2	 0	 0	 0	-1	0]
[ 0	 1	 0	 0	-2	0]
[ 0	 1	-2	-2	 0	0]
```

The resulting matrix can be used lớn determine the coefficients. In the case of a single solution, the last column of the matrix will contain the coefficients.

Simplify the result lớn get the lowest, whole integer values.

• a = 1 (MnO2)
• b = 4 (HCl)
• c = 1 (MnCl2)
• d = 1 (Cl2)
• f = 2 (H2O)

### Step 4: Substitute Coefficients and Verify Result

Count the number of atoms of each element on each side of the equation and verify that all elements and electrons (if there are charges/ions) are balanced.

MnO2 + 4 HCl = MnCl2 + Cl2 + 2 H2O

Reactants Products Mn 1 1 ✔️ 2 2 ✔️ 4 4 ✔️ 4 4 ✔️

Since there is an equal number of each element in the reactants and products of MnO2 + 4HCl = MnCl2 + Cl2 + 2H2O, the equation is balanced.