Balancing Chemical Equation HARD (do not give answer just help)

Question

I enjoy balancing chemical equations for fun but I cannot figure this out. I know I am stubborn but I refuse to look up the answer. I want to do it without "cheating" I need some help. again please DO NOT give me the answer

S + HNO₃ ---> H₂SO₄ + NO₂ + H₂0

Please No answer just hep

Every time I'm so close I'm one off for example right now I am at this:

5 S 5

28 H 28

28 N 28

84 O 84

See just one-off, and this has happened multiple times, I keep doubling the coefficient to see if it will not be a one-off but every time it ends up something like this.

I need some help on what to do next Not Answers!

Thank you so much this would really help!!

Answer 1

You might consider the algebraic method.

For: S + HNO₃ → H₂SO₄ + NO₂ + H₂O we will need to add prefix coefficients and, since we don't know what they are, we can use a variable for each:

aS + bHNO₃ → cH₂SO₄ + dNO₂ + eH₂O

where "a" is the variable representing the coefficient in front of the first sulfur, "b" represents the coefficient in front of HNO₃, etc then we can say the following:

a = c (because the sulfurs much match)

b = d (because the nitrogens must match)

So, now, to make our problem a little simpler, let's rewrite it using just "a" and "b" (substituting for "c" and "d":

aS + bHNO₃ → aH₂SO₄ + bNO₂ + eH₂O

We can now balance the hydrogens using the variables. The number of hydrogens on the left is (b)(1) and the number on the right is (a)(2) + (e)(2) so:

b = 2a + 2e

We can now balance the oxygens. On the left we have (b)(3) and on the right we have (a)(4) + (b)(2) + (e)(1) so:

3b = 4a + 2b + e and subtracting "2b" from both sides we get:

b = 4a + e

So since b = 2a + 2e and b = 4a + e then:

2a + 2e = 4a + e

or e = 2a

Now, pick a number to start from. Let's keep it simple and say a = 1

That means e = 2

Then, using b = 4a + e we can solve for b