Say you are given the equation p ( V − b ) = R T e − a R V T Where p, b, R and a are all constants.And you are asked to find an expression for d V d T , would the left side be a chain rule with a product rule with an implicit differential?So far I believe that the right hand side differentiates to: a T V e − a R V T + d V d T ∗ a V 2 e − 1 R V T Is that fully correct?

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Say you are given the equation  p  (  V  −  b  )  =  R  T      e          −              a                  R          V          T                    Where p, b, R and a are all constants.And you are asked to find an expression for             d      V              d      T      , would the left side be a chain rule with a product rule with an implicit differential?So far I believe that the right hand side differentiates to:      a          T      V            e                            −          a                          R          V          T                      +            d      V              d      T        ∗      a          V      2            e                            −          1                          R          V          T                    Is that fully correct?

Kaeden Bishop · Accepted Answer

We have  p  (  V  −  b  )  =  R  T      e          −              a                  R          V          T                          ⟹    p  d  V  =      (    R          e              −                  a                      R            V            T                                +    R    T          e              −                  a                      R            V            T                                      (              a                  R          V                      T            2                              )        )    d  T  +  R  T      e          −              a                  R          V          T                          (          a              R                  V          2                T              )    d  V  =    =      (    R          e              −                  a                      R            V            T                                +          e              −                  a                      R            V            T                                      (              a                  V          T                    )        )    d  T  +      e          −              a                  R          V          T                          (          a              V        2              )    d  Vthen            d      V              d      T        =            R              e                  −                      a                          R              V              T                                          +              a                  V          T                            e                  −                      a                          R              V              T                                                  p      −              a                  V          2                            e                  −                      a                          R              V              T                                            =            R              e                  −                      a                          R              V              T                                          +              a                  V          T                            e                  −                      a                          R              V              T                                                                    R          T                      e                          −                              a                                  R                  V                  T                                                                              V          −          b                    −              a                  V          2                            e                  −                      a                          R              V              T                                            =            R      +              a                  V          T                                              R          T                          V          −          b                    −              a                  V          2                      =    =            (      R      V      T      +      a      )      (      V      −      b      )              V        2                    (      R      T              V        2            −      a      (      V      −      b      )      )      V      T        =            (      R      V      T      +      a      )      (      V      −      b      )      V              R              T        2                    V        2            −      a      T      (      V      −      b      )

Say you are given the equation p(V−b)=RTe^(−(a)/(RVT)) Where p, b, R and a are all constants.

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