Given information:
- A B C gives product
- Doubling the volume of container, rate decreases by 8 times
- Doubling the concentration of A and reducing the concentration of B to half, rate becomes 4 times
- Increasing concentration of C by 4 times, rate increases by 64

To find: Order of the reaction

Solution:
Let's assume the rate law for the given reaction as:

Rate = k [A]^m [B]^n [C]^p

where k = rate constant and m, n, p are the order of reaction with respect to A, B, and C respectively.

Effect of doubling the volume of container:
- According to the given information, when the volume of the container is doubled, the rate decreases by 8 times.
- Let's say the initial volume of the container is V and the initial rate is R.
- When the volume is doubled to 2V, the new rate becomes R/8.
- Using the formula for the volume of a sphere, we can say that V1/V2 = (r1/r2)^3, where V1 and V2 are the volumes of the two spheres, and r1 and r2 are their radii.
- Applying this formula, we can say that (1/2)^3 = (r1/r2)^3, which gives r2 = 2r1.
- So, when the volume is doubled, the concentration of each reactant becomes half of its initial value.
- Therefore, the new rate expression becomes: (R/8) = k [A]^(m) [B]^(n) [C]^(p), where [A] = [A]0/2, [B] = [B]0/2, [C] = [C]0/2.
- Simplifying, we get: R = k [A]^(m) [B]^(n) [C]^(p)

Effect of doubling the concentration of A and reducing the concentration of B to half:
- According to the given information, when the concentration of A is doubled and the concentration of B is reduced to half, the rate becomes 4 times.
- Let's say the initial concentration of A is [A]1 and the initial concentration of B is [B]1.
- When the concentration of A is doubled to 2[A]1 and the concentration of B is reduced to [B]1/2, the new rate becomes 4R.
- Using the rate law expression, we can say that (4R) = k (2[A]1)^(m) ([B]1/2)^(n) [C]^(p).
- Simplifying, we get: R = k [A]^(2m) [B]^(n/2) [C]^(p)

Effect of increasing concentration of C by 4 times:
- According to the given information, when the concentration of C is increased by 4 times, the rate increases by 64 times.
- Let's say the initial concentration of C is [C]1.
- When the concentration of C is increased to 4[C]1, the new rate becomes 64R.
- Using the rate law expression, we can say that (64R) = k [A]^(m) [B]^(n) (4[C]1)^(p).

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