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question:How can we determine if a set of axioms is complete, in the sense that it can be used to prove all true statements about a given abstract object?
answer:In some cases, it is possible to prove completeness by showing that the axioms allow for the reduction of formulas to a canonical form and that the axioms determine the truth value of these canonical formulas. However, for complex systems like natural numbers or set theory, Gödel's incompleteness theorems demonstrate that a complete axiom system is impossible.
question:For the reaction: {eq}2CH_4(g) leftrightharpoons C_2H_2(g) + 3H_2(g) {/eq} at 2000 K, the equilibrium constant {eq}K_c {/eq} is 0.154. If a 2.0 L equilibrium mixture at this temperature contains 0.20 moles of {eq}C_2H_2(g) {/eq}, what is the equilibrium concentration of {eq}CH_4(g) {/eq}?
answer:To find the equilibrium concentration of {eq}CH_4(g) {/eq}, we can use the equilibrium constant expression: {eq}K_c=frac{[C_2H_2][H_2]^3}{[CH_4]^2} {/eq} We know that {eq}K_c=0.154 {/eq} and {eq}[C_2H_2]=0.20/2.0=0.10 M {/eq}. Substituting these values into the expression, we get: {eq}0.154=frac{(0.10)[H_2]^3}{[CH_4]^2} {/eq} Solving for {eq}[CH_4] {/eq}, we find: {eq}[CH_4]=sqrt{frac{(0.10)[H_2]^3}{0.154}} {/eq} Since the stoichiometry of the reaction shows that 3 moles of {eq}H_2 {/eq} are produced for every 2 moles of {eq}CH_4 {/eq} consumed, we have: {eq}[H_2]=frac{3}{2}[CH_4] {/eq} Substituting this into the previous equation, we get: {eq}[CH_4]=sqrt{frac{(0.10)(frac{3}{2}[CH_4])^3}{0.154}} [CH_4]=0.23 M {/eq} Therefore, the equilibrium concentration of {eq}CH_4(g) {/eq} is 0.23 M.
question:Find all real solutions to the logarithmic equation: [ frac{log (x+9)}{log (7)} + frac{log (-16x-13)}{log (7)} = frac{log (15x-4)}{log (7)} ]
answer:The given equation can be simplified by combining the logarithms since their base is the same. This yields: [ log (x+9) + log (-16x-13) = log (15x-4) ] Using the properties of logarithms, we can equate the arguments: [ (x+9)(-16x-13) = 15x-4 ] Expanding and rearranging the terms, we get a quadratic equation: [ 16x^2 + 145x + 117 = 0 ] Now, we can solve this equation for x using the quadratic formula: [ x = frac{-b pm sqrt{b^2 - 4ac}}{2a} ] where a = 16, b = 145, and c = 117. Substituting the values, we find: [ x = frac{-145 pm sqrt{145^2 - 4(16)(117)}}{2(16)} ] After calculations, we obtain the real solutions: [ x = frac{1}{8} left(-43 - sqrt{1397}right), quad x = frac{1}{8} left(-43 + sqrt{1397}right) ] Therefore, the set of all real solutions is: [ left{left{xto frac{1}{8} left(-43-sqrt{1397}right)right},left{xto frac{1}{8} left(-43+sqrt{1397}right)right}right} ]
question:What are the prime factors of the number of reasons why some people do not want a relationship with God?
answer:The answer does not specify a number of reasons, so it is impossible to determine the prime factors.